I think you are going to have to accept they didn’t marry, the note on 1939 register mentioned by Mckha489 gives a big hint that was the case. Not that unusual, and after all, if they said they were married who would query it apart from anything official? As long as there isn’t an attempt to defraud you can call yourself whatever you like, so she probably just used his surname for respectability. My aunt did exactly the same!