Is it correct to say "British English" or "English English" or simply "British" when taking about the language used in the UK?
I've seen people write English English too somewhere. Do they refer to England when they say that or UK?
You call it British English. People do that on this site all the time.
the native language of most inhabitants of England; especially : English characteristic of England and clearly distinguishable from that used elsewhere (as in the U.S. or Australia)
There are parts of Scotland, Ireland and even Wales where equating British English with the native language of most inhabitants of England could be offensive enough to get you sent to Coventry; perhaps much worse.
Even within Great Britain, let alone throughout the British Isles, the very idea that British English exists except as a hugely over-generalised contrast to however many other varieties thrive in the wild is simply risible. Even WS2 is pushing it, citing Scots as a dialect at all, let alone of English.
Many believe the best English in the world is that spoken in Edinburgh but that can be at least as different from, say, Glasgow English as Maxwell Q Klinger’s is from Charles Emerson Winchester III’s. Scots might overlap English but if it’s a dialect, that’s rather stretching the definition… in which many a Scrabble player exults.
It’s strange, today, to speak of the English of the BBC. Even 50 years ago, audiences were starting to notice first other accents and then other dialects infiltrating the BBC. Despite endless complaints and possibly even attempted counter-attacks they won, and quite easily and quickly, at least 30 years ago.