Ooh surviving the apocalypse and exploring the desolate aftermath is one of my faaaavorite
RPG past times.
Most
RPGs dealing with modern weapons would work once a suitably nuked setting was tacked on. But for actual rulesets there are a few that come to mind:
D20 Apocalypse: Basically D20 Modern (which is basically DnD) but in apocalyptic times.
Dark Future: Actually an old GW game, but focusing solely on armored cars duking it out.
Car Wars: Another car smashing game in a post apocalyptic setting.
Twilight 2000: Post WW3 game, ruleset might be a bit too technical.
There are also a bunch of new ones that I have no experience with like
Degenesis,
Hot War, etc.
Also remember that most "generic systems" have a post apocalyptic setting, so I'm sure you could find something for the GURPs system, RIFTS, FUDGE, HERO, etc.