As a matter of law, the absolute instant the King (or Queen Regnant) dies, their Heir immediately becomes King - this is the meaning of the phrase "The King is Dead. Long Live the King." and is also why the Royal Standard never flies at half staff. But as soon as is practical (normally within 24 hours) senior members of the House of Lords, after consultation with other senior figures throughout Britain, and the Commonwealth, issue the Proclamation of Accession which declares who the Heir was for clear legal purposes, and also gives the name of the new Monarch - they will have been asked what name they choose to use.
The words of this proclamation for the current Queen were as follows:
Upon the intimation that our late Most Gracious Sovereign King George the Sixth had died in his sleep at Sandringham in the early hours of this morning the Lords of the Privy Council assembled this day at St. James's Palace, and gave orders for proclaiming Her present Majesty.
WHEREAS it has pleased Almighty God to call to His Mercy our late Sovereign Lord King George the Sixth of Blessed and Glorious memory, by whose Decease the Crown is solely and rightfully come to the High and Mighty Princess Elizabeth Alexandra Mary:
WE, therefore, the Lords Spiritual and Temporal of this Realm, being here assisted with these His late Majesty's Privy Council, with representatives of other Members of the Commonwealth, with other Principal Gentlemen of Quality, with the Lord Mayor, Aldermen, and Citizens of London, do now hereby with one voice and Consent of Tongue and Heart publish and proclaim that the High and Mighty Princess Elizabeth Alexandra Mary is now, by the death of our late Sovereign of happy memory, become Queen Elizabeth the Second, by the Grace of God Queen of this Realm and of all Her other Realms and Territories, Head of the Commonwealth, Defender of the Faith, to whom Her lieges do acknowledge all Faith and constant Obedience with hearty and humble Affection, beseeching God by whom Kings and Queens do reign, to bless the Royal Princess Elizabeth the Second with long and happy Years to reign over us.
Given at St. James's Palace this Sixth Day of February in the year of our Lord one thousand nine hundred and fifty-two.
Nearly all the Monarchs of the United Kingdom have used their own first baptismal name as their regnal name - the exceptions are Queen Victoria (born Princess Alexandrina Victoria), King Edward VII (born Prince Albert Edward), and King George VI (born Prince Albert Frederick Arthur George). In the case of these last two, the stated reason they changed their name was a supposed desire by Queen Victoria that no future King use the name Albert as that would diminish the status of her beloved husband, Prince Albert - this reason is quite debatable, there is some evidence that Victoria rather wanted all future Kings to take the name Albert as well as their own in honour of her husband - "King Albert-Edward, King Albert-George, etc". Edward VII seems to have decided he didn't like that idea. So in all previous cases in the UK, where a King has not used his first name, he did use one of his other names - but he is actually free to use any name he wishes.
Very interesting. So Wills could become King William, Phillip, Louis, Arthur (or Kong or anything else). Thanks for this clarification.
I myself am a devotee of Princess Louise, that most eclectic of royals in recent history. It is said that when she visited Victoria (BC) she was so popular that there was talk of secession and making her the monarch. And that Alberta is in fact named for her. Some doubt about Alberta’s Lake Louise, however. I also thought it interesting that another lake in Canada was recently named supposedly for little Louise Windsor (Wessex).