I don't believe the births of children in American or other colonies would have been recorded at the time, in Britain, in the way you ask.
Civil registration of British births was not introduced until 1837, well after the American Revolutionary War.
Before 1837 there was a long-standing British requirement for a record to be made (at time of baptism), most usually in the infant's parish of birth.
That would imply that colonial births would most likely have been recorded in a colonial church of whatever denomination was then supported in the child's place of birth.
Children of British civil and military men serving outside Britain were customarily recorded, as dependants, in their father's service record (which may now be in The National Archives).
Sadly, I have little evidence to substantiate this view so hopefully you may get more learned contributions from others.
I wish you well