First things first:
G - The Gathering
You say that
Anyone in the world could be part of this group, therefore things like language and cultural barriers are as problematic as you'd expect.
If this is true, then those people are gathered at different times - or they arrive at different times, or there is some way that they are gathered at different times but arrive at the same time.
If people are gathered at the same time, their state in Earth previous to G will be different: if Americans are caught at work, the Chinese will be caught at sleep, and conversely. This may significantly affect the prospects of survival. If you are caught at work, or transportation, while fully awake, it may be a lot confusing to find yourself in the water of a foreign world, but you are at least awake. Being transported while sleeping will add a lot more confusion and disorientation, and might give those people a lesser chance of surviving.
If they are gathered at the same moment of the day - ie, at 8 AM in the US and 8 AM in China, and so at different times - they would arrive at different times to the new world; if the Americans arrived in the morning, the Chinese would arrive at night. And this would again massively affect their survival prospectives, those who arrived during the day having way better chances to get to the shore than those that arrived at night.
And these differences in survival opportunities between the Chinese and the Americans would probably reflect in the cultural/linguistic environment of the surviving newcomers. Randomly, about 250,000 of them should be Chinese, but if the Chinese are caught at night or arrive at night, their numbers could be severely harmed, resulting in a less heterogeneous - and less representative - sample of newcomers-who-get out from the ocean.
So the logic solution is that everybody gets caught in the morning, not too early - say, about 9:00 AM - but by some magic/fluke of reality all of them arrive simultaneously. That is, G is spread along full 24 hours, but A - The Arrival - is within a single moment.
This unfortunately does not solve a similar problem regarding the year time. If Russians or Canadians are gathered in December, they will probably be too much dressed to swim properly, and will be at a disadvantage regarding people from the Southern Hemisphere or the Tropics. So perhaps everybody needs to be disrobed for the "trip", in order to preserve the proportion between Swahili speakers and Slavic languages speakers. Or G would have to happen all across the year, with people magically arriving at the same time.
After (I suppose; with this kinds of things one should not be too assured that the arrow of time doesn't misbehave, or that causality doesn't get messed) being caught, then people will be ready for
A - The Arrival
The time of arrival won't mess with the proportions of people of different cultures/languages/nationalities as much as G - but it will affect the prospects of everybody. Few people will get to the beach if A is at night, and if people arrive late in the afternoon, even if there is still light enough for everyone to see where the shore is, they won't be able to get a decent dinner before sleep, nor a breakfast the following dawn. Their chances would be maximised, in contrast, if A was relatively early in the morning, allowing them time to collect some fruit, hunt some birds, or fish something for lunch.
Similarly, I would say pretty much everybody would die if they arrive in Winter, or late Autumn, and perhaps even early Spring, and the place of arrival is too far from the Equator. So time and place of A should be taylored for this circumstance; either something like late May at 30 degrees North, or October at 30 degrees South, or under 20 degrees from the Equator either side.
Weather will be important, too - it is different to arrive at a whole new world in a sunny day with a mild breeze, or under heavy rain and strong winds. And these waters where they are arriving - are they free from sharks and other predators? Is it the high tide? Are there strong currents?
But all of this is easy. Big problem is, we are talking about one freaking million people, and people occupy space. They won't be able to swim, or do much besides drowning in their respective places, if they do not have a considerable space to move. In a very conservative estimate, at the very least some six square meters per person, a three meter long and two meter wide rectangle, so that they can move their arms and legs without hitting and kicking each others (and I fear this is still way too crowded for a happy outcome). This means the very bare area of six million square meters, which my school teacher told me equals six square kilometers! That is, if everybody is in between 100 and 200 meters from the shore - a tenth of kilometer wide band - they need to be spread through sixty kilometers, parallel to the coast. And better the coast ahead of them be relatively plane shores, not cliffs or rocky formations. So the place for A should be sixty kilometers long of calm waters, in front of nice beaches.
So, they come to the shore. That's
L - The Landing
Now everything went well, and the best part of a million people are at the beaches of an unknown continent - or is it an island? - of an unknown world. There is again a problem of space. They will again need some 100 meters between the sea line and mountains or jungle or other rough terrain, so that they can seat down, make some protective arrangement to avoid sunburns, and be able to even think about sleeping next night.
So L needs to be situated in relatively wide coastal plains, with not too dense jungle too close to the shore, but also not completely barren, for people will need shadows to protect themselves - palm trees, perhaps, with long and wide enough leaves that one can rest under them. And people will need fresh water - rivers, brooks, lakes - unless this new world has oceans of dessalinised water. Which will quickly change the geometry of the aglomeration of people - instead of a narrow band along the seacoast, narrow bands along the riversides. Which in turn demands more depth for the plains and lightly vegetated area towards the interior.
So the placement of L is quite complicated if you want to avoid high mortality on the first day.
Now that we have a million people landed into a new, unknown world, without the pressing need of disposing with too many thousands of dead co-catastrophers, the simple phase is over. It is time for complicated things, such as
B - The Breakfast
People will soon get hungry. And besides perhaps a few candy bars or cookies they might have in their pockets, or find among the debris, they have nothing to eat. So they will look for fruit. And here is the problem; you say that this is
another planet, perfectly earth-like (same/extremely similar
flora/fauna, same size/gravity, same sun type, same moon type
And I am quite willing to believe you about gravity, sun, and moon (and atmosphere, and relief, and wheather, and whatnot), but fauna and (especially) flora will be different, barring some magical intervention from the gods who organised this event, because Earth's flora and fauna have been deeply modified by human activity. It would nice to find some bananas there, but they will probably be much smaller and less sweet than Earth's bananas, artificially selected and hybridated, during millenia, for size and flavour. And if they meet wolves or mooses, those won't be Earthan wolves or mooses, who know humans, and how much dangerous we are.
At this point, objets randomly transported from Earth may be important. Who wouldn't like to have a gun, or a steel knife, or a packet of matches (but those would be wet, and as such useless, unless they can dry them, perhaps?) in such a situation? Those things will be the more precious as they will not be renewable in the foreseeable future; once the matches have been used, fire will be only possible through more primitive methods. Luckily someone will have a magnifier, and be willing to use it to light bonfires for everyone. All hail our leader, the magnifier owner.
But let's admit that people find enough fruit - or edible mollusks and insects (though probable it will take some days starving for them to overcome their prejudices against those, and with some luck they will find ways to capture some rodents and fish, or perhaps even a deer or some other bigger prey), so B is solved. What humans do when their bellies are full? Exactly, here comes
T -The Thinking
Something extraordinary has just happened. Up to now, people had to swim, get shelter, and some food, so they were too busy to think about the extraordinariness of this event (and those who tried probably failed at the more urgent tasks). So what happened? Is this a different location on Earth? Are we going to be rescued? Why don't the cellphones work? Is it Heavens (where are my 72 virgins?!) Or is it Hell? The sun and sky seem the same as Earth's sun and sky, so why would anyoune think this is another planet? Maybe when night comes people will realise that the stars and constellations are different; if it is a night of full moon, they will noticed that it doesn't display exactly the same pattern of craters and maria we are acquainted it (how many of those million people would be able to notice that?). Or is the similarity so great that the moon is not only the same size and brightness, but has the same selenography? Is there a place in the Milky Way from where we would see essentially the same sky as in Earth?
Anyway, unless some totally unusual thing shows them that this is definitely not Earth, people will spend what part of the morning they aren't searching for food wondering about rescue, and probably wasting time at trying to bring it upon themselves. Ligthing a huge bonfire, struggling against those mute cellphones, trying to google what to do, writing on the sand, whatever. If other things, such as fauna easily recognisable as un-Terran, or differences in sky (a green sun, two moons, where is Ursa Major or the Southern Cross?) don't help their insights, they will take days, perhaps even generations, to realise that this is another planet.
This is going to affect their strategies. If you realise you are never going to see your homeland again, it is a whole other situation; you are no longer trying to get back to where you come from, and your actions need to be different. But this will be not only thought; it is going to be talked about. Now languages come into issue. You can help a person to get out of water, you can point to a fruit or for something that give some shelter, and language won't be a real problem. But you cannot discuss what this event really is unless you share a language. People who speak relatively uncommon languages will be at a disadvantage now, and only a few languages - Chinese, English, Spanish, Russian, Portuguese, Hindi, Arabic, Japanese - will have really significant amounts of speakers, in the tenths of thousands.
But with talking, comes
C - The Conflict
Now people have material for one of their preferred activities: infight. Doubtlessly, religions explanations will be immediately popular. It is the end of the world, it's the second coming of Christ, God is punishing us, we are the chosen ones (one has to wonder why how many infidels among the chosen, but who wants to get across God and his misterious ways?), on the contrary, we are the damned, repent, rejoice, prepare to die, mend your ways, pray, ask for forgiveness - there is plenty of material for big and small cults, sects, religions.
Such conflicts will have practical consequences, of much more concern than theological ones. Should we move or stay here? Should we explore the hinterland or remain close to the shoreline? Should we try to make boats and explore the sea (are there visible islands that might spark their imagination?), should we go up the river(s)?
From C, evidently, comes
D - The Dispersion
And about time, because we wouldn't like them to suffer overpopulation in an underpopulated world. They will move around, some along the shore, to the North or to the South (or East and West if it is the case), some will go up the river, some will try life in the forest or the mountains, some will get boats and try to reach other islands, some will stay close to the landing point. And this will criss-cross with the languages they speak and with the interpretations they make of the meaning of the Event. We'd rather talk in English about how God is punishing us than listen to the Japanese bragging about being the chosen ones.
Then the size of the new world lands comes into question. If it is a small island, far from the continent and from other islands, they may be condemned to an overpopulation crisis very soon. Else they will disperse and lose contact with each others; in three generations, the Event starts to become legend, the existence of other human groups may become dubious, the idea that they "came" from a different world will become more and more unlikely.
And so they prosper, multiply, and conquer the new world.
Until the next Event. Also known as Volume 8 of the Trilogy of five...
From the comments below,
However, the one thing that it's missing is technically answering the question - from here, how many generations do you believe it will take before a centralized civilization(s) forms?
Their situation somewhat reminds me of Trotsky's theory of uneven development of capitalism. These people have gadgets produced with ultimate 21st Century technology, but they do not have either the knowledge nor the material or social means to reproduce them. And, worse than that, they miss older technologies too. It is true that a lot of those people will be agricultural workers if the sample is representative of our world. But a modern agricultural worker deals with seeds that have been amply modified by a hundred centuries of continuous agriculture, and it is dubious that they can identify the wild varieties of wheat, oranges, or spinach that gave origin to the modern plants, or that they will have the knowledge of how to grow them - not to talk about how to avoid insects, fungi, and other plagues that may attack their plantations. Or how to grow any plant without additives for the soil.
They will have to try, but that experience will take months, during which people will have to resort to collection, fishing, and hunting. If they succeed, it is then possible that they will have restablished the Neolithic within a year; but that demands that people can feed themselves without agriculture for that period - without agriculture, without many modern tools and without any medieval or ancient tools (a few guns perhaps, but no bow and arrows, for instance), and without palaeolithical knowledge that has long been lost (how to make tools of silex, how to identify the signs of nearby venisom, how to fish with a spear, how to identify poisonous berries and shrooms, etc). So this is going to be a very critical and stressful first year; if they landed too far from the Equator, a failure in that first year will quite likely mean mass starvation, resulting in possible extinction or at least in greately reduced numbers. A tropical setting would be more promising because they would probably be able to survive on collected fruit and roots even during the cooler season.
If they manage to successfully grow plants in their first year, then they will have the basis for sedentary life, and some of them will be settled in villages in a relatively short span of time. It is unlikely that all that population would agree to wait for the success of the agricultural experience without dispersing into the hinterland. Once the coastal groups have amassed some grain, it is quite probable that they would be faced by those who opted for nomadism coming back and demanding some of it, which could spark a first war among them. In such scenario, the nomads, who would have been perfecting weapons for hunt, would probably be in a better position to fight. That could destroy the fledgling new civilisation, and remand all of the new world mankind into pre-agricultural modes of existence.
Of course, recognising useful knowledge and skills will be essential for survival. The problem is that knowledge and skills that are useful nowadays probably won't make any difference in such setting. What good is a IT wizard, or an engineer for people who have to learn how to hunt and how to plant? Even an artisanal smith or weaver, supposing there were some among them, what could they do without iron mines, without lambs for wool, without furnaces and looms? It is possible that people would be better served by an archaeologist who knows how to chip silex.
Worse than that, remains of modern knowledge and technology can be actually harmful. A gun and a few round of ammonition may give people some meat for some time, while delaying serious efforts to develop new weapons - and possibly allowing those who own those guns and know how to use them lopsided positions of power, from which they can actively disencourage research for technologies that would reduce their authority.
So there is going to be a huge crisis in transmission of knowledge to the next generation. Whatever those people know, they will try to pass ahead; but their ability to discern what, from their Earth experience, is useful in the new conditions will be dubious - and those born in the new world may come to reject such experience as useless and fantasist. A great problem will be writing; without proper material support for the activity - and they won't have paper, not to even talk about electronics - and without an immediately visible utility, it may be lost, or reduced to a few individuals that may make of it the basis for a caste distinction, subjugating others to the "magic" of mysterious symbols.
With those caveats, I would say that, with luck, they can be able to retain the technological level of neolithic settlements, and so start from a position corresponding to about 10 thousand years in Earthan past. They will certainly have some objects to remind them of more advanced technology, but if those things cannot be put into use and reproduced - and most of them won't - they will probably acquire the status of magic objects within three or four generations, ie, as soon as there is no longer anyone to remind them that those things were practical gadgets for their ancestors. Development from this early neolithic stage would probably be easier and faster for them than it was for us, but I wouldn't expect them to re-develop metallurgy, much less iron and steel, before several generations; they may have metallic models to inspire them, but the art of extracting metal from ore will be lost and will have to be relearnt from scratch.
Much will have to do with impredictable events - floods, droughts, rigorous winters, epidemics, that could set them back for long times and destroy accumulated knowledge. Nate Silver makes the point, in his The Signal and The Noise, that previous to the printing press, mankind could not even think of knowledge as a long term cumulative process, rather thinking defensively of knowledge as something to be preserved against the attacks of time. If so, the idea of progress, so dear to us, would be lost in the new world, because the printing press certainly would be among those things forgotten in the firts three generations.
In such a scenario, it would be difficult to predict how fast they would move ahead. Anything besides agriculture, perhaps pottery and the wheel, will quite certainly need to be reinvented, because they won't be able to rebuild it in three generations, and from the fourth generation on, the direct connection with the old world will be lost.
From the early neolithic into civilisation, we spent about 5,000 years, or 250 generations, in Earth. If they have a huge enough continent with decent rivers, or an archipel with sufficiently close and numerous islands to move from one another, if they can retain agriculture in the first years, and if they are not decimated by war or natural disaster in the first two or three generations, they should not take longer than that, and would probably be able to do it in a quite shorter time, perhaps "only" a millenium or two. But it won't be a revival of Earth's civilisations; it will be a whole new civilisation (or rather, a whole set of brand new civilisations) built from scratch.