Great, seen this debate so many times.
As for real post-apoc movies that fit into the criteria of Fallout, theres Mad Max. That's it, no others. Planet of the Apes is a close second, which F1-2 have countless references to.
Threads is a good one, the only realistic one. Made for TV movie that actually could be called a decent indie.
Six String Samurai is what you get when you just want to have fun with whacky scenarios.
Boy and his Dog: Chicken Pot Pi
(Post Second Industrial / Post Industrial Apocalypse): Which would refer to future movies in an apocalyptic setting not caused by nuclear exchange but other sources.
Alien Trilogy: Yes, it's that good. Go get the collection now.
Blade Runner: What can I say? It's a work of art. You will only get as much out of this film as you invest yourself into it. Invest yourself into it and take your time.
The Matrix: The trilogy is rather meh, but the first film had the seeds of working its way into the "art-film" category. Every single piece of spoken dialouge is a well crafted piece of pie just like Blade Runner, though not at all as good. With this film though, you get alot better visual into the landscapes and ideas that blanket this genre.
Akira: Hey, anime?!?! Not all anime sucks, just like not all cRPGs are like Baulders Gate. The actual story of the film is rather meh, but the appeal of the film is how they took a broken story and made it whole with the ability of anime artwork and style. Yep, some things actually require anime format to be expressed.
Ghost In The Shell series: About everything from this series is worth looking into. It's really really really good, and a nice world to drop into. Storyline is dry, fitting in with the context, invoking intriuge and mystery rather than lesser emotions, which at time erupt outwords in artistic fashion. Kinda like being a cyborg? Yeah, you'll want to after a few weeks. The music is beautiful, all of the artwork is beautiful, and done in a style all it's own that has been copied in so many live action movies since.
(Nuclear Apocalypse related/Pre Nuclear films)
Dr. Strangelove: This is the ultimate in sophisticated dark humor. Why? Peter Sellers and a nuclear war theme. Combine those to together and you get a film where you could laugh yourself silly dancing on oblivion. Haven't seen it? Well you suck cause this is a Fallout forum.
I'm running out of ammo, so im gonna let some others lay down suppressing fire while i GTFO and get some other suggestions.