While the accepted answer is likely the closest to the truth and the most likely initial scenario as written, it's not going to make for a very exciting story. Or rather, it will make for a different kind of story.
Note My initial reference for the countries that will spawn a dungeon will be the List of countries by population from Wikipedia and the OP's statement that the top 60 countries by population will get a dungeon. Numbers in brackets indicate their rank on the population list.
Good news: Dungeons for everyone!
TL;DR -- Best guess, maybe 6 to 24 months depending on damage done to the world in the First Wave. Expect all adventurers to be either licensed by a surviving state and/or owned by a mega-corp.
Day 0: The Initial Invasion
At this point, 60 dungeons emerge on 6 continents and a horde of entities 30 million strong spew from these portals. At this point, mana starts infusing Earth -- this point from the OP is important later.
Some will no doubt be hostile and/or hungry and/or just evil by our standards. Some might be the equivalent of refugees hoping for a better life outside the dungeon. Others still might be denizens that leave to try to keep the invasion contained. The latter two points aren't really in the OP's question, but given that the dungeons are described as a portal to another place, I will assume it a fully fleshed out world down there and not just a Murderhobo Demesne.
It is also likely that not every dungeon emerges in highly populated areas of their country unless Plot dictates otherwise. Canada's (#38) for one example is likely to emerge in a fairly uninhabited area of the country as most of their population is within about 200 kilometers of the US border. Not to say people will not be affected, just that it is less likely to effect a large population immediately unless it lands on one. Australia's (#53) will also not likely fall on a major population center as another example. Also, those invaders will have to deal with the possibly hostile and/or deadly wildlife -- Australia, I'm looking at you here for the memes.
Humour point: The first class upgrade goes not to a human but a random Australian animal, probably a snake. As if they weren't deadly enough before …
Some countries like America (#3), Bangladesh (#8), France (#20) and Italy (#23) will likely have their dungeons appear somewhere relevant and disruptive regardless of where they land in their country.
Depending on each country's detection systems and how easy the dungeons are to detect, the governments of the world could be aware in minutes.
Shortly after the invasion starts, The Twitterverse explodes with the news and first-hand accounts moments after emergence of the first portals that drop in a public area with a cellular network. Other social media platforms also explode in traffic due to this. This could bring the internet slowing to a crawl at least in the immediate term. It will have the side effect of definitely alerting the world that something is up.
How bad the world suffers for it and the overall death count will depend entirely on where in the country each dungeon shows up. As one example, the UK's (#21) dungeon appearing in London, England will be a lot more devastating to the country than it appearing in the Scottish highlands. One opening near a nuclear power plant or other electrical generation plant could be devastating in a different way.
Expect at least one international incident when a portal opens close to the border of another country that won't have one. They may or may not get along in time to gang up on the dungeon to contain it.
This is where you shape your fantasy apocalypse and will set up the rest of the story. Three primary factors will determine the initial damage suffered by the world:
- Where the portals emerge in each country
- The amount of warning that the world will get before social media alerts the world
- The immediate hostility/lethality of what escapes into the world.
Step 1: Containment
For the more militarily powerful countries and for the more readily accessible dungeons, forcible containment of their dungeon could takes days to a couple weeks as outlined in other answers that describe this process better than I could.
This is of course, dependent on said countries not being crippled by the first wave of denizens either by dropping on their capital city and killing/eating/replacing leadership, devastating key infrastructure that prevents a swift mobilization, or triggering something really nasty in their rampage.
Expect also at least one country to try something dangerously extreme and likely more than a little stupid to contain their dungeon.
However, any containment most likely can't last forever by the OP's question. As mana leaks into Earth, it is well possible that the range that the dungeon monsters are enhanced by the dungeon's mana will extend outside of the dungeon proper which means that base guns could get less and less effective as time goes on. This may also mean that the border of the dungeon moves outside of its threshold as the dungeon's mana in our world increases.
Countries fortunate enough to get a "friendlier" dungeon may be able to parley with a representative of a more peaceful faction of the dungeon dimensions for information and learn some of the rules that way and possible entreat for help against the more violent escapees.
Should the Internet survive, expect YouTube to explode in monster invasion videos as the Twitterverse and other social media calm down slightly. Expect film of the Japanese (#11) dungeon to be thought of as a live action anime adaptation of something and everyone trying to figure out what based on what has emerged.
Somebody will Instagram themselves holding a sword with a caption like "OMG! Where was that spider hiding THIS?" only with worse spelling.
The big change will be the state's First Drop. It doesn't have to be epic loot, but the moment a giant spider dies and a sword appears, it's Science Time!
For those countries forcibly containing their dungeons, this will either take the time needed to organize an initial extermination excursion or for the mana to become thick enough for the dungeon's edge to extend outside of its entrance.
Stage 2: Stand Back! We're Going to Try SCIENCE!
As stated in previous answers, this is where we as humans try to apply the scientific method to the Dungeons. Game designers might have a bit of a better time with some aspects of this given the video game feel of the Dungeon Dimensions.
Expect questions like: Would bullets made from the metal of a magically reinforced metal weapon be possible? If yes, would they still work on the denizens in the mana-rich zones? Can these Skill Books be used by more than one person? What do those bodies decompose into and can we use it somehow?
Expect a range of ethics and moralities to arise in dealing with the scientific research based on how people vie the Dungeon Dimension denizens. Excellent philosophical debates are also abound.
Also this is the part where we begin to discover what the leaking mana is doing to the environment as it is effectively foreign radiation seeping into the world. Does this do anything to our indigenous flora and fauna? Can any wolves from the dungeon interbreed with the wolves of our world and create viable offspring? Can we survive this mana exposure long-term?
Most importantly: Where did the straggling survivors of the First Wave go? Do governments have Shapeshifter Replacement Protocols? Just how many tentacles is too much for a single monster to have? It followed me home -- Can I keep it?
I would expect multiple Internet forums dedicated to the theories and physics of the Dungeon Dimensions … or lack thereof. This is, of course, if the Internet is still present and mostly public.
Any of the soldiers that might have gained a class at this point will likely be turned over to governmental scientists for study into that phenomenon, with results either publicly disseminated, kept carefully hidden, or leaked onto the Internet somewhere. Expect random stories on the internet to arise of civilians that have attained a class upgrade from the initial wave. They will be "recruited" for study -- method depends on country and its condition.
Also, theories will be abound as to who gets what class as well as Facebook/Cosmo quizzes to figure this out. They may or may not be accurate.
Initial small-scale expeditions begin with the goal to try and salvage the first minor items. Returning safely is more important at this point than loot.
Step 3: ???
At this point, we have at least some of the rules of the Dungeon Dimensions, as well as the outside generally contained barring a fluctuation in the leaking mana causing an unpredicted increase in monster resilience. Eventually there will be a transition from a purely military operation into something else if it hasn't already due to crippling in earlier stages. It is quite possible that this will be spearheaded by some multinational megacorporation for the purposed of exploitative profit.
Expeditions into the dungeons will be commissioned here and the first large-scale hauls brought up for examination and possible distribution. How and to whom they are distributed depends entirely on who owns the expeditions.
Step 4: Profit
Regardless of how it happens, the end results will be the 60 countries will likely end up exploiting the dungeons for profit, like the gold rush or the push to the explore the new world. Each country will do it in their own ways.
Dungeon towns will spring up where the containments were and soldiers will be supplemented by licensed adventurers. Those that cannot do the fighting will ply their talents elsewhere, supporting the dungeon towns, or continue on like nothing happened. I would expect a mix of surviving modern tech and fantasy trappings especially in the dungeon towns. Almost like a living renaissance faire at times for the dungeon towns.
I would also not be surprised if at least one or two dungeons were in too inhospitable of an area to be completely maintained and thus unleash a smaller wave of monsters to mop up every so often by forces and locals nearby.
It would be here that I would expect the truly adventurous to set up frontier villages in the Dungeons themselves should conditions be self-sufficient enough to permit it.
In addition to the dungeon looting, expect additional geopolitical conflict as countries without a Dungeon to exploit try to get their hands on either the loot from one or launch a force to forcibly take a dungeon. That or corporations trying to get monopolies on the dungeons.
The Rest of the World
Almost forget about the rest of the world. If the initial invasion did not kill you, destroy your livelihood, or cause you to emigrate to a safer place, life will go on as per normal most likely unless you have chosen to involve yourself in the Dungeon Craze of '19.
I would expect new niches in the workforce to open up as it would for any highly disruptive element introduced into the world. Farmers will still farm to provide food for the world and might have a surprising number of levels depending on their location. Mundane labour jobs will still be a thing, and office workers will still do paperwork. Topics might change, such as the guy whos child just hit Level 12 adventuring for Exploracorp or the executives discussing loot purchases over golf.
There will be tales of common folk heroes that have managed to kill enough denizens to level up assuming that you can at least get partial experience for those killed outside the dungeons and just not get loot.
The flora and fauna of our world will adapt and evolve to either use the released mana or to ensure they are immune to it. Neither will happen without losses first though. Nature will prove itself surprisingly resilient on this matter I would think. The Dungeon Denizens that escape into the world without being killed will provide new species for the world to adapt to as well as a new source for mythologies for our descendants.