I'm not black, so I can't speak for black people in the UK.

But in terms of English language rather than their preference, I think you use a compound term, such as Black British, it's probably more correct to capitalize, at least if you intend it to be a compound rather than intend black as "just" an adjective that happens to be used to qualify British rather than referring to a specific group. "Black" by itself would not generally be capitalized unless at the start of a sentence any more than "white" would. And this seems to be generally reflected in how I see the term used in the UK.

Thank you for the thorough explanation.