Have you seen Ada Bearpark Fulcher's naturalization form from 1908? She states she was born in London. Also, you didn't mention this but in the census in California, Anne is sometimes listed as "Suzanne" Fulcher.
This is not my family, I generally research the Fisk(e) families, as my ancestry is Robert Fiske from Suffolk, England in the 1550's.
I happen to have stumbled onto this family, out of boredom and curiosity with my own family research hitting stone walls.
If I learn anything I think will help, I'll let you know. Bob Birt