While watching the World Cup game a few minutes ago, I was wondering what if the United Kingdom - and not England - had a National Team, with English, Scottish, Welsh, and Northern Ireland players included. Is there a word we could use to refer to their nationality ? I'm not looking for the word "British". Instead I'm trying to find out whether there is a word derived from "United Kingdom".
I'm trying to find out whether there is a word [for someone born in the UK] derived from "United Kingdom".
There is no single word for a native of the United Kingdom.
There is no word in common use that is derived linguistically from the words united and kingdom and which indicates a person or persons born within the corresponding political territory.
The most common word is certainly Briton. As you note it is derived from what is now the name of a geographic territory not from the name of a current political territory.
Briton means an inhabitant of the island named Great Britain. At least some people born in Northern Ireland might not consider themselves Britons. They'd be right geographically.
You can be an inhabitant (and therefore a Briton), or even a full citizen, without being born there.
Some might argue that "United Kingdom" is a contraction of "The United Kingdom of Great Britain and Northern Ireland" and that "Briton" is therefore derived from one part of that name. That would be incorrect I think.
If this imprecision makes you unhappy, You have to use a longer phrase to express what you want.
As for national team. I'm not sure that the people concerned would all consider the UK to be a nation anyway - the word is mostly used for nations such as England, Wales, Scotland, Ireland and perhaps for subsets of at least the latter.
British citizens usually hold this status through a connection with the United Kingdom, Channel Islands and Isle of Man ("United Kingdom and Islands"). Citizens of the United Kingdom and Colonies (CUKCs) who possessed right of abode under the Immigration Act 1971 through a connection with the UK and Islands generally became British citizens on 1 January 1983. British citizenship is the most common type of British nationality, and the only one that automatically carries a right of abode in the UK.
The OED1 entry for Briton seems apposite:
A. sb. 1, A native of Britain : ... c. Since the union of England and Scotland: A native of Great Britain, or of the British Empire; much used in the 18th c.; now chiefly in poetic, rhetorical, or melodramatic use, and in phrases dating to the 'Rule Britannia' period...