Depends. Usually black if you don't know any more. Black British if you know they are British, but a lot of black people here are born in Africa or the Caribbean, and not all will be pleased to be described as British (some will take active offense, given Britains colonial past) and will prefer you to use their country or African/Caribbean depending on context.
My ex would probably grudgingly accept black British, but would describe herself as black, Nigerian, or African, despite also having British citizenship.
If you're considering how to describe someone who is present, then presumably you have a good reason and can explain the reason and ask what they prefer. If you're describing someone by appearance, 'black' is the safest most places in the UK unless you already know what they prefer.
Are extremely dark-skinned people (for example from South India) who move to england called "Black"? I've never heard that and would be surprised but i'm curious.
The term African-American does not imply that one was born in Africa. It refers to Americans of African ethnicity (which includes Carribean-Americans of African descent). Chris Rock, Lebron James, and Michael Jordan are all African-Americans born in the US.
Elon Musk is not considered African-American according to the popular usage of the term as he is of European descent despite being born in South Africa.
If they're black and British and you're describing their nationality you'd say they were British.