The family name before George V changed it to Windsor was already no longer Hanover, but rather Saxe-Coburg and Gotha, i.e. Prince Albert’s family name, as a Royal House in the UK was traditionally named in paternal line.
The situation today is somewhat inconsistent. Queen Elizabeth II has declared that the name of the Royal House should continue to be House of Windsor, but, at the same time, she said that the family name of her male-line descendants, whenever they needed or were required to use one, would be Mountbatten-Windsor, which is a composite of their paternal and maternal names ( like Habsburg-Lorraine , or Orléans e Bragança for example).
That is why , for example, the male-line descendants who are not HRHs of the first Dukes of Gloucester and Kent ( younger sons of George V) use the last name Windsor, whereas the children for example of Prince Edward , Earl of Wessex (James and Louise) have the last name Mountbatten-Windsor.
Technically, using the patrilineal criterion, the Royal House would change names when Charles became King, but it remains to be seen if Charles will acknowledge that officially, or if, as his mother decreed, the Royal House will continue to be called simply House of Windsor as opposed to Mountbatten-Windsor.