I understand that before Philip married Elizabeth, he adopted the surname “Mountbatten,” yet she announced that she and her descendants would remain in the House of Windsor. But could she have taken his name, thereby establishing the House of Mountbatten? And if she had done this, would Elizabeth’s sister Margaret and her descendants still have been considered the House of Windsor, or would they have been forced to change their surname as well?
It would seem improper of her to do so, since she’s the reigning monarch and he is the mere consort. But I suppose if she really wanted to, she could have.
Did it take an act of Parliament to get rid of that horrid German string of ancestral nomenclature, or were they able to just do it by royal decree?
From memory, George V issued a proclamation to the effect that, from then on, he and all his descendants would be members of the House of Windsor, with the personal surname - to the extent that they needed a surname - of Windsor. That’s all it took.
The present Queen has varied the situation slightly with a subsequent proclamation. She, and her descendants who are princes or princesses, are Windors; her other desdendants will have the family name Mountbatten-Windsor. Thus the daughter of Prince Edward is Lady Louise Mountbatten-Windsor
That’s what Queen Victoria did. The four Georges and William IV (and Victoria herself) were all of the House of Hanover, but her son, Edward VII, was of the House of Saxe-Coburg-Gotha, as a result of Victoria’s marriage to Prince Albert, the Prince Consort.
The alt.talk.royalty FAQ page gives a good summary, with a bit more detail than the explanation on the Queen’s own website.
If Mountbatten-Windsor is supposed to be the surname, then why are William and Harry allowed to use Wales as theirs?
Lt. Harry Wales Receives a Medal
I should think that they’d be “allowed” to use any name they chose, so long as there was no fraudulent intent.
The usual way of referring to them is “Prince William of Wales” and “Prince Henry of Wales”, not “William Windsor” or “Henry Windsor”, or even with their numerous other Christian names. Their father is not called “Charles Windsor”, and their grandmother is not called “Elizabeth Windsor”. So, if they want to use “Wales” as if it’s a surname, they really aren’t confusing anyone.
On the original question: the renaming of the House of Windsor and the resurnaming of the Duke of Edinburgh were for basically the same reason. Saxe-Coburg-Gotha was too German sounding, and the Duke’s original surname, Battenburg, was also too German sounding, around the period of the two world wars, when Britain and Germany were at war with each other. (King George V and Kaiser Wilhelm II were first cousins, but that didn’t stop them being heads of warring states).
I mean, why did the Queen allow it, if she’s already declared that when someone in the Royal Family needs a surname it will be Mountbatten-Windsor?
From the official site: The Royal Family Name
Prince Charles, and Princes William and Harry as his sons, would sign official documents with name and the differentiating element of Charles’s title, in the same way as the Most. Rev. Rowan Williams, Archbishop of Canterbury, would sign official documents “Rowan + Cant.”, signifying that he’s signing as archbishop, not as private individual, or the Spencer Churchill (Sir Winston’s cousin a couple of times removed) who is Duke of Marlborough would sign, not as “John Churchill” (or whatever his first name is, but as “John Marlborough.”
It’s worth noting, though, in terms of genealogy, that Prince Philip assumed an assumed name (and that’s not a solecism). Like this:
Back 200 years or so, one of the Grand Dukes of Hesse had a morganatic marriage, and created the offspring aristocrats under the name “Battenberg.” A son of that house ended up marrying one of Queen Victoria’s daughters and becoming a British subject, and their son was Prince Louis of Battenberg. First Sea Lord in the Admiralty until forced out by being the target of misdirected anti-German sentiment at the start of World War I. George V, who was changing his own house name from Saxe-Coburg-Gotha to Windsor, also changed Prince Louis’s surname from Battenberg to the English quasi-translation Mountbatten, and created him Marquess of Milford Haven. One of Prince Louis’s sons became Earl Mountbatten of Burma; a daughter married Prince Albert of Greece, who eas the descendant of George I of Greece. But George I was not himself a Greek by ethnic origin – he was a younger son of the King of Denmark of the time, and his family, like the Danish line and later the Norwegian line, was the House of Schleswig-Holstein-Sonderburg-Glucksburg (with umlauts sprayed about with abandon). During one of the Greek revolutions, George V sent a British flotilla to evacuate Prince Albert and his wife and son. Eventually they divorced, and the Princess, a Mountbatten by birth, entered an Orthodox convent. Their son looked to Earl Mountbatten as father figure, and when he joined the Royal Navy, renounced his Greek title and became Philip Mountbatten. A handsome Viking-looking young man in his youth, Philip attracted the attention of Princess Elizabeth, and the rest is history.
But in point of fact, the male ancestry of Philip, Charles and his siblings, and William and Harry, is Danish, the Schleswig-Holstein lineage, not Greek or Mountbatten at all.
Is that how he signed his cheques? 
No – he probably just signed them Γεώργιος 
No, he would have signed “Georgios R. Hell.” (George, King of the Hellenes) 
ETA: Damn you, Giles!! (But mine was funnier.)
Which is also why there is the tradition of British aristocrats informally using their peerage titles as the equivalent of a surname. Thus, just to take examples of peers related to the Royal Family, ‘Tony Snowdon’, ‘Patrick Lichfield’ and (before he inherited his earldom) ‘Charles Althorp’ all used those names professionally, even although none of those are/were their actual surnames.
I also have a feeling - but would be open to correction - that soldiers serving in junior ranks of the army who happen to have aristocratic titles are encouraged to do the same.
The other factor is that there is the tradition of members of the Royal Family using peerage titles as incognito surnames. This is true even when it is not an issue of wanting their identity to be secret, merely not wanting to be treated as a royal. A modern example would be the way that Prince Charles (i.e. HRH The Prince Charles Philip Arthur George, Prince of Wales…Earl of Carrick…) exhibits his paintings under the name, ‘Arthur G. Carrick’.
Properly, it should be “Cornet His Royal Highness Prince Henry of Wales” but in this instance the Queen seems to have made up her own rule (about the only thing The Queen etc. is allowed decide for themselves these days)
When the Queen served in the ATS during WWII she was known simply as “2nd Subaltern Elizabeth Windsor”, not “2nd Subaltern Her Royal Highness the Princess Elizabeth”.
Actually, “Wales” would be their title, not their surname, I believe.
It was less than that (150 years ago, I believe).
Queen Victoria’s youngest child, Princess Beatrice, married Prince Henry of Battenburg.
However, Phillip is a descendent of Queen Victoria’s third child, Princess Alice, who married Louis IV, the Grand Duke of Hesse. THEIR eldest daughter, Victoria, married Prince Louis of Battenberg, the eldest Battenberg. And THEIR son was Prince Phillip’s grandfather.
Incidentally Alice was also the mother of the last Russian Empress, Alexandra. Earl Mountbatten had a crush on his cousin Maria when they were children, and used to say he wanted to marry her when they grew up.
(I like Kaiser Wilhelm’s joke after they changed the family name to Windsor: he said he was looking forward to seeing a performance of The Merry Wives of Saxe-Coburg-Gotha)