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From the courses I've been taking, the way to say "I am British" is "저는 영국사람 이에요" which means "As for me, England person I am". But how do you distinguish between that and "I am from England but I am a Chinese person (i.e. my ethnicity is Chinese)"?
Googling for answers yielded a lot of articles about Korean ethnic nationalism, race, and identity.
Various options exist:
[ethnic origin]계 [nationality]인 e.g. 중국계 영국인. An official-sounding term, relatively neutral.
explanations involving parents / ancestry and birth. E.g. 부모님이 중국 사람인데, 저는 영국에서 태어났어요.
explanations using 오다 for the country of residence (which is assumed to be nationality) and having ethnicity as "default". E.g. 중국 사람인데, 영국에서 왔어요.
explanations involving appearance, having nationality as "default". This is actually a much more common strategy than one might expect from growing in Western cultures (at least, in British cultures). E.g. 중국인 보이지만, 연국 사람이에요. I've even heard [ethnicity]의 얼굴...
explanations involving 교포, possibly involving nationality and birth too; 국적 가지고 있는 중국 교포. Note that 교포 "defaults" to overseas Koreans, e.g. 미국 교포.
Use of 중화교포 is also quite well-understood by many Koreans of a certain age, but that has separate meanings that can get confused (either a person of Korean ethnicity with Chinese nationality, including both Mainland Chinese PRC and Taiwanese RoC nationalities; or a person of 'Chinese' ethnicity with a nationality from anywhere else).
Applicable to overseas Chinese people:
Like a lot of expressions on the border of personal identity and geopolitics, it can get complicated pretty quickly. But if you go down the explanation route, see it as a good opportunity to practise, especially with the 아주머님들!
