Pvt. William ABNEY was definitely NOT a son of Dr. Abraham ABNEY.
I have determined that Pvt. William ABNEY was a son of John ABNEY (d.bef.30 Sep 1786), son of Paul ABNEY (bapt.1699).
I wrote several articles about the erroneous connection of Abra.-Wm. You can read them in "the Abney Family Researcher" newsletters at the Centralized Abney Archives website (URL below). Of particular interest to you are: Vol.II, Iss.I, page 6 - "Parentage of Pvt. William ABNEY, et al" Vol.II, Iss.VIII, page 2 - "Paul ABNEY's 'Lost Son'"