Someone South African using the term "coloured" is different to someone British - in SA it was one of the official distinct racial categories, and didn't mean "Black" per se (there was/is also a category "Black African" on the census) but meant people usually from the Cape who had a mix of ancestry including Black, Khoi, White, San, Chinese, Malay and others.
Here it is, as others have said, usually older people, who in the best-case scenario are trying to be polite ("coloured" came in when there were a lot worse terms around) and not really getting that language has moved on. I think it's a good idea to gently tell them that other terms are better, and not to use that one any more.
Black used to be a political category here, also encompassing people of Asian descent, too (think Southall Black Sisters) but more often now I see "Black" and "Asian" used as separate categories. "BME" is useful in academic or political situations. I dislike the American terminology of "Person of Colour", not least because it implied that the model of American racial politics is applicable elsewhere, when it isn't.