told me that "British" was most often used as self-description by those who could not claim English, Scottish, Welsh or Irish* ancestry.
Yes, it's definitely going that way, though for instance - I would view the child of immigrants as English/Scottish/whatever - British would only apply to people who moved there. For instance, the well known children's author Judith Kerr is British-German to me, even though she was born in Germany and spent a good bit of her childhood there. I can't speak for England, but in Scotland, people are viewed as Scottish if they're culturally from there. On the other hand - I have a friend who was born to ME parents in the UK, and she self-identifies as British but not English. I identify her as English, though.
On the other hand, you get people like Loyalists in Northern Ireland who claim to be British, though it's hard to see them as anything other than Northern Irish, and I definitely wouldn't agree with their self-declared identity.
The census uses White British, Black British, Asian British, etc. I'd never say that some black guy who was born and brought up in London is somehow "not British', though I'd view him personally as English.