Just wondering, not trying to offend anyone.
Black people are called black. They're still British if they have British nationality. British is NOT a race. Nor is American a race.
I have never heard of white people in America being called British Americans. Where did you get that from? We are not as bothered as Americans about it and in official forms where they want to monitor ethnicity, the appropriate option for black people with British nationality is "Black British". I am White British. My German friend is White Other. My sister's partner is White Irish.
The American attitude to this is nonsensical. I knew someone who claimed to be a German-American so I spoke to him in German. He didn't understand a word. No he is NOT German, he is American of German ancestry - totally different thing.
We in the UK do not feel the need to sub-divide ourselves into anything other than English, Welsh, Scottish or Irish, because we have never had the aparthied culture that saturated (and still does) certain of the US states.
You will never hear any Normal British person refer to themselves or anyone else as "black British", "white British", West Indian British" or anything else like it.
In fact, apart from a relatively few racists (which exist in every country, even "black" African ones) we do not care about the colour of a person's skin.
What people in the United Kingdom DO care about is how people behave, regardless of their colour. This is because nobody can do anything about their skin colour, it's just an accident of birth; but everybody CAN conrtrol their personal behavioud and conduct.
This is why the UK does not tolerate certain religious and cultural practices which are common elsewhere in the world (such as public cremation of their dead, instant divorces, certain ways of killing food animals, amputations for thieves or "stonings" of adulterers, etc.).
It is nothing whatsoever to do with the colour of the people in the other countries that do these things (even some ignorant idiots in the UK think it is). It is solely because these "customs" are essentially voluntary and if someone wants to do them then they must go to a country where they are legal - not try it in the UK.
Race and Nationality are not the same thing at all. Nationality refers to your legal status as a citizen of a sovereign country; "race" refers to your ethnic origin (distinguished by physical characteristics only), which could be thousands of miles and many generations from your current position.
Neither "nationality" or "race" apply as descriptions of religion or culture in any way and should not be confused with these, which you seem to have done.
I have never heard of White people born in America being called "British Americans"
Here in the UK,we are not concerned with colour.
Someone born here is British (or if they prefer English/Scots etc)
If someone was born in America regardless of the colour of their skin they are American, why America still hangs onto putting people into race/colour/ancestral boxes is for them to explain.
Someone born in the UK is British
Black people here are called British. Never heard of British Americans.
We don't add ethnic labels to everybody. We find it amusing when we hear descriptions such as "Irish American" for someone with a vague ancestor several generations before.
The colour of you skin does not matter
Don't feed the troll.