Found it! In 1952, Queen Elizabeth II confirmed her grandfather's decision that the royal family's surname would continue to be Windsor. Her Majesty declared on 9 April 1952 that it was:
her Will and Pleasure that She and Her Children shall be styled and known as the House and Family of Windsor, and that Her descendants other than female descendants who marry and their descendants shall bear the name of Windsor.
A few years later, HM The Queen modified this statement by issuing Letters Patent in February 1960 which stated in part:
while I and my children will continue to be styled and known as the House and Family of Windsor, my descendants, other than descendants enjoying the style, title or attributes of Royal Highness and the titular dignity of Prince or Princess, and female descendants who marry and their descendants, shall bear the name Mountbatten-Windsor.
Did this mean that the name of some members of the royal family changed from "Windsor" to "Mountbatten-Windsor"? Some people contend that the goal of this declaration was meant to not only change the surname of the children of HM The Queen but those of her male-line descendants as well. At Princess Anne's wedding in November 1974, Anne signed the marriage register 'Anne', without a surname. It was the registrar who filled in her names as 'Anne Elizabeth Alice Louise Mountbatten-Windsor'. According to a Buckingham Palace statement issued in October 1975, the specific addition of the surname 'Mountbatten-Windsor' was "the Queen's decision that this should be done". Further, HM The Queen consulted with the acting Prime Minister to confirm whether all her children would have the surname Mountbatten-Windsor. She received the following reply:
"The effect of Your Majesty's Declaration is that all the children of Your Majesty who may at any time need a surname have the surnames of Mountbatten-Windsor."
(Prince Philip: A Biography, by Denis Judd, London: Michael John, 1980, page 196)
It would seem that the surname of HM The Queen's children is whatever HM wishes. Legally and constitutionally, however, the Queen cannot do as she wishes. The surname of the Queen's children is Mountbatten-Windsor in practise and has appeared three times: at Princess Anne's first marriage in 1974, on Prince Andrew's marriage register in 1986, and when the banns were read prior to Princess Anne's second marriage to Commander Laurence in 1992. (When the Prince of Wales married in 1982, he signed the register as "Charles P" and the registrar filled in his name as "His Royal Highness Prince Charles Philip Arthur George The Prince of Wales".) Nonetheless, the family name remains legally Windsor because there hasn't been any modification or clarification to the Letters Patent of 1960.