Fantasy worlds are usually set in a separate world, time, and space from our own. Many are set in an alternate world and dimension. They feature medieval-like setting and technology.
Yet some fantasy stories are set in our world. Imagine that an apocalyptic catastrophe changes our world into the medieval-like fantasy setting. Is this solution valid? Does it make sense?
You can set a story anywhere. The challenge is not to make it consistent with our world but to make it self-consistent within itself. And I think this is a universal literary problem (and therefore I don't think this question belongs on worldbuilding). All stories take place in what Tolkien called a sub-created universe. Even the most gritty realism is a fabrication that takes places in a simpler, neater, less random universe than our own.
For example, the kinds of random chance and coincidence that happen in real life all the time do not work in stories. Readers feel that these things are cheats, that they violate story rules. Story worlds, even without magic, obey their own rules of causality and probability.
So, you can construct a world such as you describe. You simply need to make it a self-consistent story world. It needs to have its rules, they need to be consistent with each other, they have to accord with the general rules of story worlds, and you have to stick to them.
But these are rules about shape, not about content. The reason that you see so many fantasy stories set in pseudo-medieval worlds is that it is easier for authors to borrow these existing tropes than do the work to create something new, and readers have less work to do as well, because the rules of these worlds are already well known to them. Establishing a new trope is much harder work, but it is not restricted in any way by the rules of existing tropes. It is restricted only by the rules of story.
The fantasy genre can apply to a large range of settings. So yes, and I think you already answered your question, a post-apocalyptic setting can fit, as long as you make it believable to your readers.
But I think that your question would be more appropriate for the Worldbuilding stack exchange
Edit : And although your setting may be post-apocalyptic it doesn't mean that it should be the main focus of your story. You can make it a seemingly regular fantasy world but drop some faint hints that it is a post-ap world.
Done well, this can work -- but getting it to work well is tricky (as I'm finding.) IMHO, one of the trickiest aspects is to provide a plausible reason that both big changes (tech dies; magic appears) happen at the same time. Why did those both happen? Did something science-y/technical in a lab somewhere cause a world-wide (or universe-wide?) change? Have the stars finally aligned just right to awaken Lovecraftian Old Ones?
In science, “Extraordinary claims require extraordinary evidence.” (--Carl Sagan.)
In fiction, we have more wiggle room than that, but your readers will want an explanation that makes sense (to them), within the premises of your story. The setup/world design needs care, so your readers will buy into your reason why guns, cellphones and cars (for instance) no longer work.
A previous answerer/commenter mentioned Steven Stirling's Emberverse series (there are two loosely-linked sets of novels.) If you haven't read Dies the Fire, IMHO you're in for a treat! That said, I found his explanation a bit troubling. His unseen "alien space bats" did this to earth, but we don't know anything about why they bothered.
Give the readers a plausible reason for the change, and you've got a great starting canvas to work on. Lots of problems/conflicts both from the old ways no longer working and the new (or even Older, if you will) ways coming back and mucking things up for your characters.
P.S: I've seen a few questions about this basic idea on the worldbuilding stackexchange and they've been allowed if posed narrowly enough that people can answer; come see!
You may take a look at "The daughters of the Dragons" (Die Töchter des Drachen) by Wolfgang Hohlbein. While it is a fantasy story, with dragons and magic it is after a third of the book revealed it takes place in a future version of earth, after it was once destroyed.
So, it is your world, if you want swords and magic in your world, then let there be swords and magic in your world.
Magic, for example, may be the result of a genetic mutation caused by nuclear fallout after the big war.
You're ideas are totally fine.
Seriously, as a reader I would have no idea with a disaster turning us all into cool, fantasy characters, ready to go and... Be fantasy characters. In fact, it'd be pretty cool if that actually happened. However, I think I'd like to add these notes to the slew of answers this question has accumulated:
They probably wouldn't think 'yay!', rather... Something else. One reason it's hard to make this sort of stuff believable is because the public would react so strangely.
I think it's just so hard to make this realistic and believable. It's miles easier to just have the change happen before the book (create a prologue perhaps) or have no change at all. In fact, yeah, I think you could have a prologue saying the change then the book. It's a bit hard to write about the characters adapting because it's just... It's just so weird.
There's no problems with these ideas, they might be a bit hard to write about.
Other's may disagree, but in my experience, doing things like this is tough. So I wish you luck. You're ideas are doable and viable, and I'd love to hear how they turn out.
I hope this helped you.
