Ten did say in TDOTD "I'm gonna be King!", but more likely he would have been given a dukedom and styled "Royal Highness".
In English/Irish history:
Geoffrey Plantagenet, Count of Anjou never received any titles from his wife when she was "Lady of the English" (though that post was disputed anyway).
Lord Guildford Dudley wanted to be made King when Jane took the crown (also disputed). She said he would be king "by me, by Act of Parliament". Later she said he would not be king at all, but she would create him Duke of Clarence. He managed to convince ambassadors from Brussels to call him "King Guildford".
Philip II of Spain was made a co-monarch with Mary I. An Act of Parliament gave him the title of King. In all official documents, and on coins, the couple were shown together. Philip lost these titles when Mary passed away.
In Scottish history, Margaret had no spouse, Mary I had three:
Francis II of France was King Consort of Scots between his wife's accession and his own death.
Henry Stuart was "
His Grace The King of Scots" throughout his marriage. He was also created Duke of Albany eight days before the wedding.
James Hepburn, 4th Earl of Bothwell was created Duke of Orkney and Marquess of Fife on 12th May 1567, but all his peerages were removed by Act of Parliament on December 29th. He never held the title of King, nor any royal style.
In English/Scottish/Irish combined history:
William & Mary were crowned together as co-regents (it was sometimes called a Diarchy).William held the majority of power (really he had won the throne by conquest), and continued to reign alone after his wife died.
Prince George of Denmark and Norway married Anne on 18 July 1683, He was created Duke of Cumberland, Earl of Kendal and Baron Wokingham by William III six years later, taking his seat in the House of Lords on 20 April 1689. When his wife became Queen, he was content to remain the dutiful consort and he never made a bid for the title King.
In UK and British Empire/Commonwealth history:
Prince Albert of Saxe-Coburg & Gotha married Victoria in 1840, but was not actually created "Prince Consort" until 1857. The government refused to elevate him further, though he eventually amassed enough political influence that Charles Greville noted "He is King to all intents and purposes.".
Prince Philip of Greece and Denmark renounced his foreign tiles to become a British citizen (though this later turned out to be
unnecessary). On 20th November 1947 (just before his wedding), George VI created Sir Philip Mountbatten 1st Duke of Edinburgh, Earl of Merioneth and Baron Greenwich. On their tenth wedding anniversary, Elizabeth II created him "The Prince Philip".
So the Doctor may have been King of England and Ireland, or he may have only been made a Duke (this assumes that nobody questioned why the Queen was marrying an oddly dressed nobody in a wooden blue booth with a flashing lantern on the top).
With regards to Sir John, it is noteworthy that none of the other Doctors have so much as an MBE between them (perhaps the Queen is worried that if she knights any one of them then all the others will start demanding gongs as well).
The real fun starts if any of them get a life peerage. (I can imagine other peers taking the piss during the introduction ceremony, maybe making the inductee wear a Gallifreyan collar on top of the ermine robe.)
