I would never use *Great* Britain - which is sheer pretention. The only exception being the abbreviation GB, and then only where it is well known, as in the phrase 'GB number plates'. The Communist Party of Great Britain suffered endless ridicule for that concession to national chauvinism.
The United Kingdom is the formal name of the political entity that falls under the sovereignty of the parliament at Westminster. So strictly as a description of the legal jurisdiction, as with the post, UK is acceptable.
As to the political nation, the truth is that Britain is a fair description. The Union of 1707 is reasonably well enduring, and it would not be true to say that a specifically English imperialism is responsible for the domination of much of the World in the 19C. - the Scots especially and to a lesser extent the Welsh have made their contribution to the martial culture of what is properly called British imperialism. The reputation of the Scottish regiments in the North of Ireland is quite as low as that of the English.
The Scottish and Welsh object to the elision of English with British, on the grounds that Britain is more than England, and will pull you up for saying English when you mean British. So, it is wrong to say, for example, that the critique of English political economy was the basis of Marx's Capital, when two of its most important exponents were Scots (Smith and Hume) and another Welsh (Richard Jones).
However, when referring to relations between the nations of Britain, obviously, you are talking about relations between England, Wales and Scotland. At which point, one might pejoratively talk of English imperialism (towards the other two). Also, 'English Culture' does capture a specifically philistine streak in the national psyche.
The special case is Northern Ireland, the Six Counties, Ulster. Unlike the union with England and Wales, the union with Ireland was decisively rejected. The partition of Ireland is an artificial attempt to corral a majority of protestants loyal to the Crown, by drawing the boundary to suit that effect. Loyalists and Tories call it Ulster, which is a fiction: Of the nine counties of Ulster only six are included in the jurisdiction of the UK. The other three are in the Irish Republic (or Eire), because if all of Ulster was in the UK it would contain a mojority of catholics, who would vote for union with the Republic.
More accurately nationalist (that's Irish nationalists) call it northern Ireland (with the 'northern' uncapitalised), when referring to the geographical area, or 'the Six Counties' when referring to the polity. That way they avoid giving credence to the fiction of the 'United Kingdom of Great Britain and Northern Ireland', with its fanciful myth of a 'Union' between Britain and the wholly artificial orange state of _N_orthern Ireland.
Hardline republicans also refuse to acknowledge the legitimacy of the government of the South, referring to it derogatorily as 'the Free State' (the name that the supporters of the Treaty with Britain in 1922 gave it), or Eire, but not the Irish Republic, which according to republicanism, was snuffed out with the defeat of the all-Ireland Dail (parliament) of 1918.
-- Jim heartfield