The main problem you're gonna have with trying to find a pre-existing set for a "nuclear apocalypse" is that the idea behind a nuclear apocalypse is "everything has been destroyed". Intact buildings or empty fields or whatever just wouldn't fit at all. Unless you're willing to burn down a forest, I suggest taking Gonzo's idea and setting it inside a room. If you can, find a place that looks like it's not been used for a while - so an old, abandoned building. There are safety and legal issues with that though. The best approach would probably be to just get a normal room, somewhat dark lighting and set it up to look like some sort of shelter.
Gonzo actually made a fantastic short film called
The Island, which, whilst not based in a "nuclear apocalypse", did do a fantastic job of showing the story of a character who had secluded himself from society after the island had been invaded by... something. At least I think that's what the story was, it's been a while since I've watched it. The basic idea of a potential "end of the world" scenario is there though.
Also, Wayside Creations made a Fallout based short film recently that's being made into a web series. It's called
Fallout: Nuka Break, and the backstory of the Fallout series is essentially a nuclear apocalypse, just like what you're aiming for. What they did for a location was, I think, go to some sort of desert. They however had that nearby, and had a much bigger budget.
The main problem you're going to come across in the UK is there's hardly any, if any at all, locations you could shoot a "nuclear apocalypse". It's best to aim for something within your own limits, especially when you're an independent film maker. Not everyone can pull a Nolan and tell people to do everything without post-production effects (I was watching
The Dark Knight behind the scenes stuff yesterday - my god that man is insane).
I suggest working on other things until you can think of some way to properly do this. If you don't think the room idea of Gonzo's is good enough, then you're going to have a very tough job on your hands getting this to look right.
tl;dr: Go with Gonzo's advice, or perhaps work on something else until a "nuclear apocalypse" setting is within your limits.