As he uses the title 'Prince' he does not use a surname other than (to this point) Wales. If he was any other male member of the family NOT using the title Prince it is apparentl Mountbatten-Windsor, per the Queens's request, although formally just Windsor
He actually doesn't have a surname but for any purpose if him needing one for some reason it's Mountbatton-Windsor. Although on his RAF uniform his name badge says Wales. And I know that because I was the one that started the thread asking about it recently.
Rather a long while ago, the queen decreed that her male descendants would bear the surname "Windsor" and the female ones would be "Mountbatten-Windsor". I do not know if this has been changed.