Apocalypse World is a pretty good one. It’s very rules-lite and narrative focused. Players are more in control of the narrative than most RPGs; the players can declare what they want to do (e.g. “I want to drive really fast towards that group of bandits, then jump out of the truck just before it hits them, do a backflip in the air, and shoot the two on the left while the truck runs over the two on the right”), and rather than the response being “You can’t do that”, it’s typically “Okay, but”. Getting a die roll that isn’t high enough to just flat out accomplish what you were trying typically still lets you succeed, but with some caveats, which the player typically gets to make the final decision on - for instance, in the above case, you might roll a 7 (a middle-tier roll), and be told that you can choose 2 of the following: You accurately shoot the two bandits on the left, the truck kills the two on the right, you don’t hurt yourself when you land, and your gun doesn’t explode.

Flatpak

Flatpack: Fix the Future is the first game in the smash Hyperoptimistic Postapocalyptic genre of gaming. Flatpack is a game about building a new society using pre-apocalypse technology that you don’t always trust or understand, and solving terrible problems creatively. 

Sorcery and Super Science might work for this. It’s already most of the way there. Most characters are mutants though and all humans are sorcerers so it might not be a perfect fit

You can also try a bit of a combo move with Worlds Without Number and Stars Without Number. Both games could probably handle this by themselves, but the rules are similar enough to combine which would give you scifi and magic together

I think Barbarians of the Aftermath could be a good fit for you. It allows to make up your own apocalypse, to customize it as you like. And it based on the simple and excellent Barbarians of Lemuria system.

I can think of three options, none are perfect, but each could be tinkered with to get what you want:

In the Light of a Ghost Star – This is much further in the future than you intend. But it has incredibly simple rules which are easy to customize. Also I made a psionics supplement for it here. I had pretty good success running it.

Mutants & Machine-guns - A loose, fairly tongue-in-cheek system. Free.

Mutant Future - Basically a cleaned up clone of early edition Gamma World. Less light than other games mentioned.

How simple do you want to get? You could reduce PbtA down to a single move Defy Danger. Still works.

Mutants & Masterminds 3e – available for free online as d20herosrd.com – can do literally anything. It’s originally written for superheroes (obviously), but I’ve seen it used for D&D style fantasy, sci-fi, and so on. There is literally one single spell in all of fiction and film that I have tried to replicate in that system and not been able to (Balefire, from The Wheel of Time, because weird shit with time). It’s all the flexibility of GURPS with a fraction as much math, all the power-fantasy of D&D without the game-breaking power-loops (there are a few but they’re easy to spot and avoid). And all the complexity is front-loaded; once you get through character creation, it’s all just d20s. If you can play D&D you can play M&M.

For something less crunchy, try Fate. Whether Core or Accelerated or whatever I couldn’t tell you – I personally hate this entire family of systems – but it is very rules-light and storytelling-focused. It might suit your needs.