I agree with eric (#13) and would only add that to the best of my knowledge, 'Red Indians' was only ever BE, and is very possibly now outdated. It actually sounds rather childish to my AE ears, maybe because children in old-fashioned BE books played at Red Indians, while children in America would play cowboys and Indians.
The usual term in AE is just 'Indian,' then later more specifically 'American Indian' in formal contexts, and now more politically correctly 'Native American.' Of course, 'Native American' and 'African-American' are both usually used mainly in formal/academic/bureaucratic contexts. They're much less common in everyday speech, where the normal words are still 'Indian' and 'black,' which aren't dated or derogatory in a casual, everyday context, only in more formal writing.
Of course in English, neither 'Indian' nor 'American' is imprecise enough to matter, as the meaning is almost always obvious from context. If you need to describe Indians from Central and South America, you just add an explanatory phrase like that, or say 'South American Indians,' or use a paraphrase like 'indigenous peoples of South America,' or use specific names like Aztec, Maya, Inca, Aymara, etc.
I haven't ever heard 'First Nations' used by anyone except Canadians either; I took it to be a Canadianism.