Thanks for clarifying a bit.
I think although Philip didn't specifically request the King of Greece to relinquish his titles, it would appear that by becoming a naturalized British citizen and taking the name Lt. Philip Mountbatten, he would in effect relinquish his titles in the eyes of the British law. Whether he would still have his titles in the eyes of the Greek law I don't know.
According to British law, it doesn't appear that he was naturalized under the Act of Naturalization for the Descendents of Electress Sophie. The British Home Office has a quite informative page on the law:
http://www.ind.homeoffice.gov.uk/ind/en/home/laws___policy/policy_instructions/nis/e-i/hanover__electress.html
Prince Ernst August of Hannover claimed citizenship under the act but the page doesn't mention Prince Philip. Futhermore, from this page, I gather that the rights of citizenship under the act wouldn't have given Philip the right to live in Britain. That wouldn't have helped him much.
It does appear that whether or not Philip had royal titles himself, they would not have made much difference to his children's titles because their mother was heiress to the throne. The only other Queen who married a member of the reigning house and had children was Queen Anne who married Prince George of Denmark. They had several children who died in infancy but one who lived to adulthood was simply styled, HRH Duke of Gloucester.
It appears that the confusion on Prince Philip's title at his marriage result from two things:
1. Husbands, unlike wives, don't automatically assume the titles of their royal wives,
2. Children generally inherit their titles from their fathers, not their mothers.
3. As stated above, George V didn't take into consideration the titles of the children of a female heir in his Letters of Patent. If he had, there would have been less confusion.
I think another source of confusion is what people refer to when they say upon marriage. Philip's status upon marriage would generally be considered his status he brought to the union when it was approved, not the status that the King conferred on him after approving the marriage. Yes, technically Philip was a Royal Highness at his marriage but that was only because George VI created him one in anticipation of the marriage. But the status that he brought into a proposed marriage was of a British commoner subject of royal ancestry.
So it appears that Philip had during his lifetime three statuses:
From birth to naturalization: Prince Philip of Greece (royal status-foreign)
From naturalization to the approval of the marriage: Lt. Philip Mountbatten (non royal status-British subject-commoner)
From approval of the marriage to late fifties, HRH Duke of Edinburgh (royal status-British) and thereafter Prince of the UK (royal status-British)