I thought most people would say which part of the UK they are from: England, Scotland, Wales or NI?

Yeah that happens too. I generally say UK. In Europe I’m maybe prone to saying England/English perhaps, like in Italy “Io sono Inglese” but in the UK my parents are welsh and I was born in England so don’t really identify with either very strongly. Particularly because I’m a scouser, like you’d identify yourself to other British people by your accent if you have a strong obvious one. Regardless though nobody says “GB”.