The Royal Family Name according to the Royals. (I snipped out a lot)
Before 1917, members of the British Royal Family had no surname, but only the name of the house or dynasty to which they belonged.
In 1917, there was a radical change, when George V specifically adopted Windsor, not only as the name of the 'House' or dynasty, but also as the surname of his family. The family name was changed as a result of anti-German feeling during the First World War, and the name Windsor was adopted after the Castle of the same name.
... in 1960, The Queen and The Duke of Edinburgh decided that they would like their own direct descendants to be distinguished from the rest of the Royal Family (without changing the name of the Royal House), as Windsor is the surname used by all the male and unmarried female descendants of George V.
It was therefore declared in the Privy Council that The Queen's descendants, other than those with the style of Royal Highness and the title of Prince/Princess, or female descendants who marry, would carry the name of Mountbatten-Windsor.
This reflected Prince Philip's surname. In 1947, when Prince Philip of Greece became naturalised, he assumed the name of Philip Mountbatten as a Lieutenant in the Royal Navy.
The effect of the declaration was that all The Queen's children, on occasions when they needed a surname, would have the surname Mountbatten-Windsor.
AND
>Prince William's full title is His Royal Highness Prince William of Wales KG. [The Queen made him a Knight of the Garter in 2008]
In his military life, Prince William is known as Flight Lieutenant William Wales.
Edited to add: HIs father is
His Royal Highness Prince Charles Philip Arthur George, Prince of Wales, KG, KT, GCB, OM, AK, QSO, PC, ADC, Earl of Chester, Duke of Cornwall, Duke of Rothesay, Earl of Carrick, Baron of Renfrew, Lord of the Isles and Prince and Great Steward of Scotland.
Edited by: The Rt. Hon. Christopher Geidt CVO OBE, Private Secretary to the Sovereign