To trace a child in this sort if circumstances it is very important to understand some of the rules around how birth indexes work.
Ultimately, you can't be sure of the marital status of the parents(s) nor even whether both are named with 100% certainty just from the index entry alone, but you can sometimes narrow the possibilities down a little ....if you know the following:
The child has no surname itself on a birth register entry (before 1969), so you need to remember that the surnames in the index are those of one or both of the parents named on the entry, depending on their marital status.
So it is never a question of a woman "giving" a child a previous husbands surname - if she is still using that surname herself, then that is the name the entry will be indexed under, as appears to be the case for the 1945 registration.
A maiden name appearing in the index, does not necessarily mean that a father is named on the entry or that the parents are married.
Nor (for registration purposes) is a woman's husband automatically deemed to be the father of her child - she CAN name him as such without him being present, but if she does that knowing it to be false then she commits perjury.
She subsequently had a second child (my father) who was born in 1945. His birth was original registered under her married name with father unknown which was subsequently updated in 1947 when she married my grandfather.
It wouldn't say unknown, it would be blank - an unmarried father can't be entered on the birth register entry unless he is present to sign the register with the mother as a joint informant.
The 1947 entry will be a re-registration (under the Legitimacy Act) after the parents marriage.