If in doubt, say Britain and British (never 'Britisher' which is a made up word only Americans use). The country as a whole is Great Britain, or the United Kingdom of Great Britain and Northern Ireland to give it it's full name. The constituent parts are England, Scotland, Wales, Northern Ireland and a few other small things. Very few people will object to being called British, whereas if you call a Scot, Irishman or Welshman 'English' or refer to the whole country as 'England' by mistake, they will hate you for ever. Think of it like the difference between the US and individual US states, except that England by virtue of being larger and the seat of government tends to dominate the others.

American politics are, IMO, completely broken. I'm looking for things that work -- or at least, things that are better, even if they're flawed in different ways.
Do you mean your entire political system and parts of the underlying constitution, or those people currently practising politics and hence the way the populace at large responds to them?