I think I would normally use the names I grew up learning: England, Scotland, Ireland, Wales, Northern Ireland. (I imagine some of these aren't quite countries but they're the divisions I think of when I think of the UK.) I would say London, England, for example, but I've seen London, UK recently and I'm still not quite used to it. I would use British for the general term but I wouldn't think normally of calling someone from Scotland British. I suppose it's correct but it seems odd to me. They live in the UK but they're always Scottish to me. The same holds true for Northern Ireland. I think of them as Irish, not British, although I imagine they have British passports and would go to the British Embassy if they had a problem. In political discussions I tend of think of it as the UK because it's one government that is acting.