Not much help, but I did find Lena and daughter Gertrude in the 1940 census.
Ridgewood, Bergen Co., NJ:
Lena Bradshaw, age 39, b. VA, widow, Negro (the term African-American was not used in those days), servant
Rutherford, Bergen Co., NJ:
Gertrude Stanton, age 22, b. VA, single, Negro, servant
Also, I did find an Alexander Bradshaw still in Virginia in 1940, age 12, Negro, a grandson in the household of James E. Hamlin, age 69, and wife Lucy, age 67. They had a son Simmie, age 17. Also grandsons James Randolph, age 10 and Dorsie Randolph, age 6. The location was Prince Edward Co.
I then looked for James E. Hamlin in 1920 (didn't find him) and 1910. In 1910 he is in Prince Edward Co., and has several children, but none named Lena. There is a daughter Sarah, age 9, which is the age Lena would be, but don't know if this is her.
Sorry I couldn't find more.