You may need to consider the possibility that she was not entirely truthful on the marriage certificate. It was clearly quite a rushed job. Having looked at the groom's service records, he was probably on embarkation leave when he married. There is no mention of the marriage in there - I wonder if he needed permission from his senior officer and failed to obtain it? He certainly doesn't seem to have bothered to amend his next of kin, which is odd. There is some correspondence about his will. You say that his family were unaware of the marriage. This may be true, but there is also the possibility that they DID know but kept quiet, as the will would have been rendered void by his marriage.
Although the marriage was in London, I doubt if Elsie was from there. The service record puts him in camps on Salisbury Plain for the whole of his stay in the UK, so a Wiltshire lass is more likely.
Caroline