My answer is going to take the question to the best-case extreme. We have had lots of questions of this type on Worldbuilding before, but they usually take a form where people can only make the trip once and have to bring everything they need up front. By allowing endless trips back and forth at will we can always have plenty of experts and skilled workers in every field and plenty of help. Disallowing non-living items also puts an interesting spin on this.
TL;DR: On the order of 1-10 years if it just pops in and enough skilled people agree to try and rush-build a new modern society there and they coordinate well. Or on the order of 1 month - 1 year if we know it's coming and can prepare and practice for it and have a colossal level of support and expertise from thousands or millions of people.
Assumptions:
- The area on the other side will have reasonably close access to all necessary resources.
- Concerning only living matter passes: I'm going to bring plants, but only if they are fully mature plants still alive enough that you could plant them in the ground and they could keep growing, that should count as alive.
- Any potential hold-ups such as "What is this thing? Is it dangerous? Let's study it first" have all happened and are not counted in this answer. The answer starts at "We know what's going on, let's do it now and quick."
This answer will follow two different cases as I go along.
Case #1: Earth evacuation. Maybe we find out aliens are going to invade Earth sometime soon, or whatever. We have a fire under our butts. Most of the people on Earth are willing to support the effort, and millions of people give their time and skills.
Case #2: Gameworld. The portal was designed and created by the most advanced of the ET races we have met, and they created it as a huge game. Each time the game is started the portal shows up on the participating race's home-worlds. You have advance notice about when and where the portal will appear, and you have time to prepare. We have a line of people waiting at the portal location who know exactly what their jobs are and what they plan to do each day. As soon as the portal appears, the single-file line can rush in and everyone can hurry to their required activities. It is the largest-scale multiplayer game in the history of the universe. Everything happens as fast as it possibly can, only taking as long as it takes to physically perform the minimal amount of actions necessary for the fastest method of reaching the goal.
For many of the actions in the gameworld case, imagine groups of people who have practiced working together to perform one type of task as fast as possible, like race car pit crews who can work together flawlessly to do several things to a car and have it driving away all in seconds. And they will have lots of teams making lots of everything and cutting corners and hurrying things; it's ok if most things fail or break early on - as long as we have hundreds of teams rushing the same thing a few of them should succeed.
Let's get building!
Go!
The first thing we need to do is set up our foundation.
Starting with primitive tools
Fortunately there are crazy awesome people who create primitive tools from natural resources just for fun or research who we can use as guidelines. Go check out the YouTube channel Primitive Technology if you haven't already done so.
Some of what follows is also from personal experience. I consider myself to be a hobby bush-crafter, or hobby "primitive survival" practitioner. I've made stone tools, started fires with flint and steel and with friction (ie: "rubbing sticks together"), made rope from plants, fired pottery the old fashioned way, dug garden beds with a stick, and done other things you typically associate with this type of activity.
Hand Tools
As long as you have various kinds of rocks you can make some very crude stone tools useful for a few tasks in less than 1 hour. If you get experienced flint nappers in your group who find flint, you could have sharp knives and axes very soon. Also, if we are allowed to bring an animal across such as a beaver and then extract its tooth, those can make good carving tools so you could have one almost immediately, but I'll just assume you wait a day for the good flint tools. By the way, flint tools can be made sharper than iron tools, they just have other drawbacks.
Earth evacuation hand tools: 1-2 days
Gameworld hand tools: A few hours
Clothing
As soon as sharp tools are available make clothes out of animal hides. If you skin it and wash it you can start wearing it almost immediately. It might seem gross, but I'd rather be wearing a hide that still smells like its previous owner than to keep going naked. We'll want better clothes later, but this is sufficient for now.
Both cases: First day
Fire
With good fire starting materials, even natural ones grabbed from the wild, the best fire starters can walk out into the wilderness, grab what they need, and have a fire on day 1 if there are some very dry materials available. It could take a few days or weeks if the materials need some processing such as drying in the sun or carving with the stone tools.
Earth evacuation: 1 day to 1 week
Gameworld: First day. Even if it's not dry enough, with enough muscle power from lots of people we can dry things out by friction.
Manufacturing
Very primitive manufacturing was done by shaping wood or making pottery. With the right people, you'll be chopping down wood and carving it on your very first day here. Pottery will take longer since you have to find a good clay source, shape it, dry it, then fire it.
You can also create water resistant objects using only wood fire ash and water. You wet the ashes, make a thick goop out of them, and use it similar to clay. It does not even need to be fired, you just leave it out for days to dry. I just learned this yesterday from one of the newer Primitive Technology videos where the guy does it and shows that an ash-bowl filled with water still has half the water left the next day.
Earth evacuation: Wood manufacturing will start on day 1 or 2. Clay firing will start after a few days for small items if you have fire by then, up to a few weeks for larger items that take longer for the clay to dry.
Gameworld: Wood manufacturing first day. Clay firing can take a shortcut: with enough help we can create so many clay objects that we don't care if most of them fail to fire, just fire them after a day or two and hope 1 in 1000 works. For larger clay objects you'll probably have to wait at least a few days. Time to finished pottery obects: a day or two for small objects, a few days to a week for large ones.
Farming
Someone can start working on a garden bed immediately on day 1. Even if you want to tear up the ground (not necessary for all farming techniques) you can still do it with nothing more than a stick you pick up or break from a tree.
Both cases: Food is available on day 1, some gardening is done and mature plants in the ground on day 1, and food is not really a problem.
Build Time: Primitive Technology
Before long you will have:
- huts for shelter, a shed full of knives, axes, adzes, rope, and other simple tools.
- a garden bed full of plants transplanted from Earth
- whatever they want made of clay including plates, bowls, bricks, tiles, etc.
- stone hearths and fireplaces
Earth evacuation: Less than a month
Gameworld: A week
Metal Age
They will have already started on the building blocks for more advanced technologies before they complete the "primitive" section above. Someone will already have started working on a clay furnace and forge. Someone else will already have started working on a bellows. Others have already been searching for ores. I'm just calling it the metal age since they can start on them all at the same time, even iron. And I'm going to concentrate on iron here.
As soon as 1 or 2 furnaces are ready, someone will already have found at least a few rocks with ores in them. Even if it takes a few tries, there will be smelted metal in less than a month, especially if a bellows is completed in time.
The first metalworking tools are sticks and stones, so the first tools made out of metal will have lots of impurities and be very crudely made, but they can function. The first iron pokers, hammers, and knives on wooden handles will be hastily made all still inside that first month.
Using those tools, better tools will be made. These tools will include all of the conventional blacksmithing tools and some knives, axes, adzes, saws, horseshoes, barrel rings, wheel bindings.
Yes, even horseshoes and wheel bindings: we might as well get the beasts of burden over here now if they aren't already there. Make sure you've been working on carts and carriages in the meantime.
Now you have homes, gardens, metal hand tools, flocks of animals, vehicles like carriages and carts, barrels, and lots of other things. Since the heating technology is already well advanced at this point you could make glass too as long as you've acquired the materials for it.
Evacuation: 1-2 months
Gameworld: 1-2 weeks
Manufacturing Age
At the end of the previous section, you've already had everything needed to start making the first woodworking lathes. Most of it can be made out of wood, and the cutting edge can be as simple as a blade or a sharp point. I have done rough lathe work before by putting a stick in a drill, attaching something to the stick to spin it, and applying a knife held in my other hand to the wood. For our group, they don't have the hand drill, but they could make something very similar.
The first woodworking lathes could come before you even have your iron work going since you could do it with flint cutting tools. But once you have iron tools it will be even more reliable.
Now you will have lots of lathed wood objects, and as soon as a blacksmith makes an iron drill bit you can get a woodworking mill going too, so your mass manufacturing base has essentially started.
What you can not do well at this time is mill or lathe iron objects, but you can use iron mill and lathe cutting tools on some other metals like copper, tin, or lead. And you'll have various other tools at this point for rolling metal, bending metal, etc..
Your metal industry is mostly limited by how much metal can be acquired. If you can get lots of it, this world will have transformed into something that looks like Earth in the 17-or-1800s. Lots of things could be mass manufactured at this point, even non-iron metal objects.
At this time iron objects, including steel which is made from iron, will probably still be made on an anvil until you can make cutting tools of steel-cutting quality.
Both: Many tool bodies can be made of wood and already crafted and waiting for their cutting edges ahead of time. So this point should be only days or weeks after the metal working starts.
Chemistry
We have all this cool manufacturing, blacksmithing, glass, etc., but to continue making this colony into something resembling modern Earth we need chemistry.
We're not just now starting on the chemistry, instead the work should already have been ongoing since the beginning. Ever since we had fires, primitive chemists will have been starting to create lots of chemicals. Once there were clay vials, they can be working on lots of liquid chemicals too.
Before the end of the previous section, there will already have been gasifiers, beakers, tubes, and various other tools created and we'll be at a chemistry level of technology similar to the 1800s or early 1900s, except that we'll only have relatively small quantities of the chemicals so there will be an effort to simply mass produce as much as we can before we can actually start using it to do much of interest.
Two of the more useful things that might come out of this are fuel and aluminum. Fuel might come from gasified wood or from plant ethanol. Aluminum will be made using the Bayer/Hall-Heroult Processes.
Aluminum is important because it has a much lower melting point than iron, it makes a fine conductor (though not as good as copper), and it can be worked on easier than steal. Also the ground is full of tons of the ingredients for aluminum.
Another big one to try and push for soon would be plastics and rubbers. Then we can start making covers for wiring to get ready for our later electronics. Plastics and rubbers will also make it easier to do chemistry on gases in addition to the liquids and solids we've already been working with.
With our manufacturing in full swing, aluminum available to compliment our blacksmithing steal work, and fuel from plants, we might be able to start working on some automotive prototypes, the first ones probably wood frames with the simplest engine we can make.
A lot of the chemistry initial buildup will happen at the same time as the previous sections, but some will have to wait until we're manufacturing stuff. We might have some prototypes for some of these things likes automotives and wiring soon, but we would have to create a factory before we could mass produce these things. The initial versions will be crude and potentially dangerous, but they could be rushed into existence around this time.
Both: Most of this was done concurrently with previous sections. Some will have to wait until after and will take some days to months to build up various useful chemical stocks.
Electricity
It might sound like a bit of a stretch, but it would not be unreasonable to start generating some electricity as soon as we have conductors that we can start connecting fruits and vegetables with. Most people have heard of using citrus fruits as power sources, but you can do it with other thing too such as potatoes. As long as you bring a still-live one over, so it goes through the portal successfully, you can use it as soon as you can wire them up. In some stores you can even buy kits that power things this way. My kids had a potato-powered clock kit at one time.
Barring plant-power, we can start making batteries if we find the necessary materials. If you have the correct materials at hand, you can make a battery just by piling the right materials together in the right way.
Both of the previous two electricity-generating methods could be done as soon as you can make the wires.
As soon as we have wires and magnets we can start making electricity the way we actually do it in power plants. Once we get to this point, we can set up larger scale power generating plants. We could do this soon after arriving as long as we have magnets. To actually do much of interest with the electricity will require the manufacturing from the previous section.
Evacuation: Some months
Gameworld: Some weeks
1900s level
We're already building devices to use with our electrical grid and we have manufacturing and chemistry. How long it takes to have various useful devices depends on their complexity.
Some examples of things you could expect rough versions of:
- electric motors/engines
- lights
- heaters, stoves
- radios
- air-tight containers
- airplane prototypes (early Wright brothers level)
- 1800s style "submarines" (they are not what you imagine when you say submarine)
- rocket prototypes
- 1800s or early 1900s style automotives
- pumps (both water and air)
- telegraphs and maybe telephones
By this point you should not expect:
- electronic computers, at least not useful ones yet
- TVs, monitors
- useful rockets
Evacuation: Less than a year
Gameworld: Some months
We might be able to have some form of crude 8-bit electronic computer soon. It would probably be just a few of them built, they would be designed for their specific purpose, and they would be used only to help bootstrap the rest of the computer revolution. These initial ones would be slow, mouse-less, monitor-less things whose time would be very valuable until they were used to help design and create the next generation of computers. Remember, this is being assisted directly by Earth computers and engineers on the other side of the portal, we only need the ones here just to get some form of automation going and then that world's computing revolution can explode.
Since we are assuming limited previous life on this planet there might be no major source of fossil fuels. That is just as well: it may speed the development along, but it's all the better that we start right out developing renewable, sustainable, cleaner energy sources. It would probably start with wind and water first, and we might be trying to make solar panels after that.
Evacuation: I think we would see prototypes and some more functional thing we associate with 1900s level technology before the end of the second year.
Gameworld: Gameworld is still rushing headlong getting everything done literally as fast as humanly possible. We're probably still not approaching the end of the first year yet.
2000s level
We can get up to 1800s and some early 1900s level by cutting a lot of corners. But going that last step to having very precise machining and manufacturing of precision parts and having microscopic-scale manufacturing for computer parts is going to be one of the trickier parts.
How long this would take is a hot topic of debate among some people. I feel the research is inconclusive.
It certainly should not take decades though. It took decades on Earth. If we have the push for this that I described initially, we should be able to do it many times faster than it took us on Earth. After all, we already have it and understand it, we're just recreating it, and we're doing so with all experts readily available.
So going from the end of the previous section to being fully 2000's style modern should take on the order of years to redo it, 1 year for Gameworlders who know exactly what to do and just do it. I would say months, but there are activities which need to be physically done, tested, possibly redone, such as physically creating computers, rockets, planes, etc.. It's not just a matter of drawing the design up in your first computer you make.
Conclusion
All of the initial setup checkpoints will take days, weeks, or months. Then to get to the level of a few hundred years ago will take months, setting up mass-production (relative to what is already there) will take months.
Then progress will slow down as you approach modern Earth level and you try to make everything very precise, very large scale mass production, build up a large quantity of essential components, and recreate complicated structures like modern rockets, modern cars and planes, and modern computers, medical equipment, etc..
The entire process from day 1 to the day you could almost mistake it for Earth today would be on the rough order of...
Evacuation: 1-10 years
Gameworld: 1 month to 1 year
This is the best case scenario for fastest time, and it is borderline crazy but might technically be possible, especially since there is advance notice and preparation, and maybe even practice runs.
Now I feel like I want to start a club who actually does this type of thing. What government will let us do all this in their border?