The games are great, I'd even say almost perfect. Except the time limits. I can't stand a game where I only have so little time to explore and do whatever. I just want to explore the wasteland and discover things on my own, but unlike the later games where the main plot can wait for as long as it needs until you're ready... here I am forced to do the main plot and nothing else, because doing anything else takes up precious time.

Am I to assume, once I finish the main plot, I can just wander and do whatever? Or do I get a "The End" and have to start over?

I think you're missing the point: choice and consequence. You do not have to rush to do the main plot. You may take your time, and the associated consequences. Choice and consequence.

If you send the water caravan to the vault you'll get an extention.

I get it now! In turn, one cannot do EVERY quest. I can only do some, as the others are for lack of a better word, doomed. This is the consequences and in turn creates a realistic world where I'm not some superbeing who solves everyone's problems. Most are doomed to suffer.

This also in turn encourages multiple plays in which I decide "what if I did that instead?"

You can't do every quest because some become closed when others open, but, you can do essentially a "complete" playthrough, not missing sidequests unless you close them due to choices, AND complete the main quest on time, if you know where you're going and what to do and don't kill a bunch of time traveling and resting.

True, the same for the later Fallouts as well, though hilariously if you make your intelligence as low as possible (at least in 3 and New Vegas) quite a few NPCs refuse to even give you quests because you're such a moron (I wouldn't trust a complete idiot with solving my problems either)

Exactly!

The time limit is only really an issue in the first Fallout, where you have to get the water chip, and that time limit can be significantly extended in the Hub. Then it will be really tough to run out the time limit unless you spend weeks upon weeks aimlessly wandering the wasteland looking for random or special encounters.
In any case, the game doesn't end when you finish the water chip quest. The game continues beyond that point without a time limit. About 1/4 or so of the game's content is meant for after you've finished the water chip quest.

And Fallout 2 doesn't have a time limit at all from what I can recall. At least not one that's anything to worry about. I seem to recall reading that technically there is something like a 5000 day limit or something.

Fallout 2 has a time limit, 13 in-game years.
Which is a LOOOOOOOT.
I had to cheat with Sfall (you can change the speed of the game), and use the sleep option tons of time to skip years in pipboy.
You can see the game over screen in YouTube:
https://www.youtube.com/watch?v=qwzFJzedzfU
Playing normally, I never got out of time, and I finished the game dozens of times.

Woot! Then I'm not far out then with my 5000 days :D But yeah, that is an insane time limit

It's insane! :)
I heard you can reset the time limit, but I never needed so I don't know if it works perfectly.
It involves changing a line in Sfall config file (see link below).

http://fallout.answers.wikia.com/wiki/Time_limit_elimination_-_Fallout_2

Really, I was just playing like usual, when the Shaman's face appears "You're running out of time!"

Then later "You're running out of more time!"

I read that he only does this three times, and then after some time my village collapses. I wonder what the hell was I doing all that time because I don't recall getting anything actually accomplished.

This are just small cinematics of the game, to remind you that the survival of the village is in your hands.
You can ignore Hakunin messages completely and play for 13 years.
You have all the time in the world, don't worry!
You can't solve the village problems before his last telepathic message, so there's no reason to hurry. ;)

Unlike the cinematics of Fallout 1. If you ignore them, you'll run out of time quite fast.
After you solve the water problem, you can play more relaxed.

Thanks :) the constant warnings really made me feel pressured. I mean, the game itself really seemed to love to rush me against the clock, even if that wasn't the case it felt like it.

No problem ^^
Just be careful that some sidequests have a time limit.
Like the quest Jo gives to you, in Modoc.
So if you're trying to do a perfect/near perfect run, getting good ending for the cities, etc, save a lot!
Better to be safe than sorry.