britain = great britain = the UK = which doesn't have a capital

London= capital of ENGLAND, england=a part of the UK.

or..?

There's no differenc ebetween Britain and Great Britain. GBP referse to our currency "great british pound", but we hardly use it otherwise.
UK is Britain- the United Kingdom of Britain (though I don't know why we are still a Kingdom and not a Queendom, given that we have a female monarch on the thrown)

NO.
Britain has three capitals- London of England, Cardiff of Wales and Edinbeurgh of Scotland.
If you want to go by the full title, "Her Majesty's United Kingdom of Britsin and Northern Ireland", you can include Dublin of Northern Ireland, too.
But there is no overall capital like America has.