My husband's great-grandparents came to the USA in 1898, and returned in 1904. While they were in the USA, they had two girls, one born in 1898, the other in 1900. On the 1911 Census (England), the girls were listed as British Subjects born in the US, like this:
It's difficult to read, but I think it says "Brit. Sub by Parentage".
Would their parents have been required to register their births, or might there be any other paperwork re-asserting their status as British subjects? Would they have been required to renounce their American citizenship?
The older girl died in 1915 before reaching majority, and the younger, as far as I can tell, married another British Subject in 1919 or 1920.
(The parents of this family did not become naturalized citizens while they were in the US.)