What is a British person called?
England is called Anglia. British people in general are called brit or in plural britek but the term is less widespread. Great Britain is called Nagy-Britannia but the United Kingdom is called Egyesült Királyság.
Is there a word for people from the UK?
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.
What do you call someone from London?
A person from London is known as a Londoner. City of London.
What do you call a British girl?
Bird. This is British slang for a girl or a woman.