a. In 1960, she had her first child born out of wedlock.
a1. In 1960, she had her first child who was born out of wedlock.
Do those imply that she had had other children before 1960 who were not born out of wedlock?
I think in theory at least it is possible that her first child born out of wedlock was also her first child.
But it would be clearer to say:
b. In 1960, she had her first child**,** who was born out of wedlock.
In short, could one use (a) and (a1) instead of (b)?
Your sentences A and A1 mean exactly the same thing because A is an abbreviated form of A1. Sentence B has a different meaning.
Both sentences mean that she had other children out of wedlock after that one. They say nothing about whether she had any other children in wedlock, either before or after that one in 1960.
Your sentence B is clear that her first child was born out of wedlock in 1960, and that she had other children. It says nothing about whether any other children were born out of wedlock.
A indeed implies that she had children before that, or the author would probably not emphasize the fact that this baby was a first (but just how "she had a baby out of wedlock").
A1 is unclear. By omitting a comma (between 'child' and 'who') it can be interpreted both ways, but likely still implies this was not her first child in general.
It's unlikely the author would write such a sentence (in English) and not catch on to that ambivalence.
Your version, B, is clear in that regard.
(As a sidenote, she could technically have given birth to a child earlier in 1960, and not only before that :)
