I don't mind individual quests with time limits, as long as those time limits are only for how long I have to complete them once I start them, and not something silly like "How long I have to start them from when I start the game" (i.e. A quest must be started and finished within the third month from the beginning of the game). What I really hate is how Fallout and 2 have time limits to finish the game, even though Fallout's limit can be extended (and then shortened again, if certain events occur). I like to have time to dink around and find all the little hidden stuff and Easter eggs and all.

I know that quite a few endgame states in Fallout 1 are dependent on time limits such as:

*SPOILERS*
Running out of time to find a water chip, obviously.
Super Mutants overrunning Vault 13 if you let slip its location.
*END SPOILERS*

But if I want to see these endings, I can look them up on YouTube. Is the time limit disabled by default in the GOG version, or are there custom frontends that do that for me?

The time limit is still in Fallout.

I think there was a time limit in Fo2 that was patched out. Sorry, I don't have all the information on that.

PS: Technically there's a limit in Fo2, but it's like years and years. The only time I've heard of someone reaching it is on purpose.

So where are the patches that get rid of the time limits?

Oh, the GOG versions are all up to date. There's still the water chip time limit in the first game... if there was ever a patch or mod for that, it would have been unofficial. Maybe someone made one. FO2 is also up to date, so that should be fine.

Besides time limits, though, Fallout 2 does have one problem which, for me, required a patch. You travel on the world map, right? Well, with today's CPU's, you go way too fast, and have way too many random encounters. Really annoying. So you need Killaps patch.
http://www.killap.net/fallout2/patches/unofficialFO2patch.exe

Ok, so where are the mods that get rid of the time limits?

You could do a lot worst than check out No Mutants Allowed for a variety of fallout patches/mods
http://www.nma-fallout.com/forum/dload.php?action=category&cat_id=3 is a direct link to the Fallout 1 files section of that estimable website, if it's been made (and I don't doubt that it does) it's in one of those files

I don't see anything on either of those links that relates directly to removing the time limit.

One google search led me to this site:
http://uk.answers.yahoo.com/question/index?qid=20081028105636AAHru2z

Which again leads to the site which Matchstickman has mentions above.
Reading the FAQ at that page will reveal that the Fallout patch 1.1 removes the second
time limit (the GOG version should already contain this patch).
The water chip time limit (which is 150 days) is not removed.

Note.
Several other sites mention that there isn't a modification for removing the first time limit.
Keep in mind that it's really big.

EDIT.
Quote from the FAQ:

Hm, yeah. I would have thought someone would have made one.

It's not that big of a thing. You'll be able to level up enough that getting the water chip won't be difficult within the time limit, and after that you can just explore as much as you want without any limits.

Yeah, and I pointed you to the site where it will be.
Look around the site and if you can't find it there ask on the forum on that site, they will know.

Yeah, and I told you I looked around that site. Stow the 'tude, queenie. Anyway, I'm getting a "Server not found" error from every link I try to enter the site from.

Dude, no 'tude.
Just reminding you that there's a dedicated community for Fallout that is far more likely to know the exact mod/patch you want than GoG is.

And yes, the server does appear to be down right now, first time I've seen that on NMA in a long while (certainly the first time this year).

Even the links from their Twitter page (which makes no mention of any downtime) give the same error.

Yeah when the server is down every link should give the same error. Maybe they haven't noticed or hadn't time to fix it yet, as they are not payed for their Fallout Fan page and are probably at work or something.
Have patience and try again in a few hours or tomorrow. As it is the biggest remaing Fallout Fan site I know of you will probably only find mod/patch links the lead to them, so searching for other sources for the timie limit removal patch could be futile.
As far as I remember there was a patch that remove the time limit removal fan patch, but I don't have it lying around because I haven't used one yet.