I always find it hilarious when Americans talk about English accents and seem to think there are one - or maybe two if they've seen any period movies or Mary Poppins -, given there are several clearly distinct English accents in use in my London borough alone (ignoring accents with immigrant origin, which would add many more)

- Hey can you tell me where "lie-sester" square is?

- Oh you mean "lester" square, yeah walk up that...

- No I'm pretty sure it's "lie-sester"

- Ok well I've never heard of that square, good luck!

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.