Discussion here https://www.ancestry.com/boards/thread.aspx?o=0&m=2808.3.2.1.1.... suggests that Hannah Deming was actually married to a *James* Bird, and that Andrew's wife was named Mary.
While his father may have been *a* John Bird, it is unlikely in the extreme that he was the *same* John Bird who married Grace Stegge. Children of *that* pair went straight to Virginia and didn't mess around with the North.