Google Ngram Viewer allows you to count the relative occurrence of a word or phrase in the Google Books library for a given year of publication.
From 1820 to 2000, 'coloured people', 'colored people', 'black people', 'aboriginal' for all books published in English. You can see the great rise in 'black' and beginning of the decline in 'colo(u)red' (previously the polite usage for what we now call black) in the late 1960s, coterminous with the post-Civil Rights rhetoric of Black Consciousness and Black Power.
But if you do the same only for books published in the UK, you'll see that 'black' only took off in the 1980s, a good twenty years after the States. That poor officer whose use of the word 'coloured' was held up by the Lawrence Enquiry as evidence of police racism wasn't terribly far behind the publishing curve.