The slightly tricky answer to this question is Merle Oberon who was an Anglo Indian. There are quite a few current British/Asian actresses who have become fairly famous but I doubt if any other became a big star in Hollywood. She married Alexander Korda and became Lady Korda.
I think some people tried, years ago, to imply that an even more famous star, Vivien Leigh, was Anglo Indian but I think that was nonsense.