ONLY if BHO Sr. and SAD were legally married.
In the common law, and all statutes in effect in 1961, bastards did not HAVE fathers.
The concept of legal fathers for bastards was invented by the US Supreme Court in 1973 in Gomez v. Perez (the same Court in the same term that decided Roe v. Wade).
BHO Sr. could not have transmitted British nationality to any children of SAD, since they were not legally married.
Which speaks directly to my last, which see.
Furthermore: If it was the Supreme Court that invented the idea of fatherhood for the propagators of bastards in a holding of the Court, then that is case law not constitutional law and is subject to revision and review by any subsequent session of the Court, or any act of the Congress.
It also would arguably not apply to The Won because he was born in 1961.
It also would not modify the black-letter law spelled out in Article II.