It appears to be a late registration ( although I'd want to see the actual entry to be more certain). I can see no likely original registration in 1907, although it's always possible a registration could have been done under some pretend/married name which would make it difficult to trace. If it was a re-registration you would usually expect a marginal note saying "previously registered on ....." or something similar ( although they don't always get noted that way).
It isn't a re-registration under the Legitimacy Act, because that hadn't been introduced in 1923.
From the details on the certificate given before - it appears to be a simple late registration giving details of a birth to an unmarried woman (but who has since married). As she appears on the entry with two surnames, the entry is indexed under both Gadsby and Bywater.
There is nothing to indicate Mr Bywater is the father. If he was - he could have been named on the entry as the father if he had attended the registration and signed as a joint informant, which he obviously didn't do.
Having gone to the trouble of getting authority for a such a late registration, which would have needed an application to the Registrar General, then if he had been the father, I would have expected the couple to have made sure he did get his name on the entry as such.
Late registrations or re-registrations are never connected with any sort of adoption, or step parent being added.
