I've never tried it myself, but have thought about it as well. If you're thinking ground ball screws, forget about it, leave it to the pros. For cheap rolled screws it might be doable, but probably not worthwhile.
It should be possible to anneal the part of the screw you need to cut. Wrap the screw in a wet rag where you don't want to cut it, and heat the part you do want to cut til dull red then let it cool on its own.
With the right carbide inserts you can just cut through the hardened stuff. It's probably just case hardened, so you won't have to cut too much tough stuff.
You could even try grinding through the hard skin.
You're probably going to need more than a £500 lathe to do a decent job though. Typically in that price range lathes just don't have the rigidity to take accurate cuts in steel. I think a 10" swing would be a bare minimum.