Hi David
I hope you don't take this the wrong way, but I think your conclusions about William's navy service need to be more rigorously tested. I say this because on the NGS medal roll the William on HMS Vanguard is a ships cook, but later on in HMS Albion is a Leading Seaman, which would be an unusual career path in the Royal Navy. If you were able to visit the National Archives at Kew I would consult Albion's Ships Description Book (ADM 38/7489), this would confirm this William's age and place of birth, plus listing all his previous ships and dates of service.
There is another path you could take, and that is to order on-line from the National Archives, the Greenwich Hospital School admissions application for John Charles Blackmore, born 5th March 1841 (ADM 73/172/87). While this does not provide a place of birth it does give his parents as William and Sarah. A subsequent check on Familysearch.org returned exactly the same birth date for a John Charles Blackmore born to a William and Sarah in Wiltshire.
Admission to the GHS was only available to children of long serving seamen, and applications were generally accompanied by a statement of the man's service. Of course this still might not be your man, but my hunch is that it could well be.
Martin