Traditionally, if the King dies, his wife if she is still alive becomes known as the Queen Mother - as she is the mother not the spouse of the new monarch.
If Queen Elizabeth dies and Prince Philip is still alive does he get any special title like the Prince Father to reflect that he is the living parent of the new monarch?
Short answer: not unless they/parliament felt so inclined.
One note is that traditionally a king is expected to be head of the royal household, so a king regnant will name his wife a queen consort, but a queen regnant will name her husband a prince consort in order that she’s not outranked by her groom.
So the naming of a queen as “queen mother” (or “queen dowager”, etc.) is to separate her from the reigning queen, but there are expected to be any number of princes. He’d keep all his titles, but as prince rather than prince consort.
It’s not traditional for the queen mother to be known formally as The Queen Mother. It was a one-time deal because “Queen Elizabeth” was ambiguous. Queen Mary was factually the queen mother, but she was never Queen Mary The Queen Mother like her daughter-in-law was. She was just Queen Mary.
He’s a prince in his own right, not just as a consort. His status is not directly related to his marriage. Obviously it’s because of who he married, but there is no legal link between his titles and his marriage to the Queen. If he and the Queen were to divorce, nothing would change automatically just because of the divorce. Of course, his being HRH and a prince are things that are at the whim of the monarch, so she could take those away at any time. (She can’t stop him from being the Duke of Edinburgh, though.)
Prince Phillip does not have the title of Prince Consort currently - only Prince Albert had that. He is plain old Prince Philip, Duke of Edinburgh. His titles would not change as they are not directly linked to his marriage.
As others have noted, there’s nothing traditional about the ‘Queen Mother’ title. When baby George ascends to the thrown then his mother, if she’s still alive, will be plain old Queen Catherine. Well, unless George also marries a Kate, I suppose.
Also, to nitpick a lot, that would only the case if she’s actually the mother of the new monarch. Sometimes she’s the grandmother or aunt or stepmother.
When William IV died in 1837, the dowager queen Adelaide was the aunt of the new queen regnant Victoria.
Widower, presumably
Not quite, as it was not entirely unknown for Queen Mary to be referred to as ‘the Queen Mother’ in official contexts during her sons’ reigns - she was ‘Mary the Queen Mother’ in the prayers for the Royal Family in the Book of Common Prayer between 1936 and 1952, following the pattern used for the previous queen dowagers who were the mothers of the reigning monarch, Henrietta Maria and Alexandra.
A previous Philip, married to the sister of a previous Elizabeth ( to whom he later proposed after Mary’s death — no mid-Victorian qualms about marrying a deceased wife’s sister for him ) ruled England jointly with his wife as King Consort, much as did William and Mary ( the last being the formal claimant to stealing her father’s throne, and Billy Boy not wanting to be seen as less important than his wretched wife ). Afterwards he went back to being King of Spain etc. etc. and making war on everyone he possibly could, including the Pope.
A more similar situation to the present people would be Henry Darnley, King Consort of Scotland, whose main achievement was restoring the Stuart surname to the House of Stuart ( e.g. had Mary of Scots married a Guise or a Hamilton, she would have taken her husband’s surname ); but who due to a rare moment of intelligence in his wife was precluded from sharing the royal power.
However he did not outlive her.
There were however two crucial differences between Philip of Spain and William of Orange. The 1554 Act for the marriage of Mary and Philip specifically granted Philip the ‘kingly name’ only during Mary’s lifetime and implied that he could only advise her; the 1689 Bill of Rights, in contrast, specifically allowed William to continue as king if Mary predeceased him, as well as giving William alone ‘the sole and full exercise of the regal power’ even during her lifetime. In terms of what really mattered, the differences between them were more important than the superficial similarity.
APB, my recollection is that Claverhouse does not accept the validity of the Convention Parliament, so citing the 1689 Bill of Rights to him is an invalid authority. 
Philip will continue to be known as ‘rude bigot’. 
Now that a first-born female can ascend to the throne just as well as a first-born male, one might wonder if the popular sensibilities on the above topic will shift towards being more gender-neutral as well. Perhaps by the next time there is a queen regnant, there will be no qualm about naming her husband the King Consort.
(ETA: Or, (gasp) maybe there will someday be a King Regnant and his King Consort!
:eek: 
)
I have to wonder whether this constitutes political snark in GQ? From a moderation perspective, I mean. Obviously, it is, but it’s hard to take it so grimly seriously when it’s about a political event three centuries gone by. 