Somehow I don't think D&D would work for a relatively realistic apocalyptic campaign. I'm thinking it should be set soon after the apocalypse, like a decade or so, or even during it, rather than post-apocalyptic set centuries after the apocalypse. What do you think?

My personal favorite system is Fate Condensed. It's a universal system with relatively light rules and is super customizable. It puts story way above mechanics, though, which may be an issue for a lot of DnD players.

No that's fine. I've been looking for slightly simpler alternatives to D&D.

It sounds interesting. I hope the things you mentioned will be added. It will be an ultimate fantasy world simulation game.

There is one that would work fairly well, 

The Fallout TTG system.

That or a homebrewed Cyberpunk system.

Quote from TheFlamboyantGnome >>

Somehow I don't think D&D would work for a relatively realistic apocalyptic campaign. I'm thinking it should be set soon after the apocalypse, like a decade or so, or even during it, rather than post-apocalyptic set centuries after the apocalypse. What do you think?

 Can you give a bit more detail on what exactly the apocalypse would be? I have a few post-apocalypse homebrews I've played. I could see what they have, but I wanna make sure it would fit the same theme.

Free League's "Year Zero Engine" (YZE) games use very simple dice pool mechanics.  Some use straight pools of d6s, others have easy to comprehend use of different dice d6-d12.  Attributes and skills are on a easy to pick up D through A scale.

The YZE exists under an OGL so you can see the basic mechanics laid out for free, and they've actually incorporated some of the YZE individual games into the OGL too, so you can get a pretty good preview of how the system works before you invest in a particular game or just homebrew it.  The first YZE games were actually for an apocalyptic setting, starting with Mutant Year Zero (everyone is some sort of human "mutant" in an apocalyptic scavenger scape, it's a bit like the classic Gamma World, but a little more down to earth).  That was followed up  by Gen Lab Alpha (where all the characters are "uplifted" and anthropomorphized animal species escaped from laboratories to contend with the same apocalypse as M:YZ ... it's sorta like Palladium's After the Bomb, but a little tighter).  Elysium is the book for unmutated humans in the world, who live a sort of Fall Out type siloed existence.  There's a robot book that's out of print, and supposedly "Ad Astra" will come out some day which takes the game to stars. It's fun if you want a little far fetched ness to your apocalypse (psionic powers and lasers).

You just missed the kickstarter for early access to Free League's Walking Dead licensed game, but that'll be out by the end of the year I believe.  Same YZE system applied to Walking Dead, and from what I understand while Zombies will be a threat to the game, more of the mechanics will focus on simple survival, community building, and negotiating with other communities.  There's also Free League's go at a 4th ed of Twilight: 2000 set after a "WWIII that never was" that's also a strong game if you want your post apocalypse play to have a sort of military/war bent.

Basically Free League puts out consistently good product around a very fun, easy to learn and incredibly flexible mechanics engine.  They also do Bladerunner, Alien, Tales from the Loop, and an in house IP called Vaesen that's sort of like the TV show "Grim" meets CoC of sorts, Coriolus which is sort of space faring apocalypse, and Forbidden Lands which is their fantasy game that's much more old school in terms of grit, exploration, and surival themes in a fantasy setting.  You can set up the challenge to something like Tales from the Loop where the worst thing that happens to child characters is they get scared and run away in terror, to very lethal games where not thinking through situations or failing to take an abundance of caution will get PCs killed quick.  

I'd check it out.