Aren't they actually Mountbatten-Windsors or are they going to use the name Cambridge as a surname?
LaRae
The family name is 'Mountbatten-Windsor' for anyone who needs a surname.
HRHs don't.
They use the 'title' of the title holder as a sort of surname to distinguish which branch of the family they are from so
'of York' - Elizabeth until her father became King George VI
'of Edinburgh' - Charles and Anne until their mother became Elizabeth II
'of Wales' - William until he married and Harry at the moment
'of Cambridge' - George and Charlotte
'of York' - Beatrice and Eugenie - probably until they marry when the 'of York will be dropped'
'of Gloucester' - Richard until he succeeded to the Dukedom
'of Kent' - Edward, until he succeeded to the Dukedom, Michael to this day and Alexandra until she married when she dropped the 'of Kent'
George and Charlotte, assuming that things proceed as expected will use the following names:
'of Cambridge'
'of Cornwall and Cambridge' - precedent - the children of George V and Queen Mary from the death of Queen Victoria until 9th November 1901 when George was created Prince of Wales
'of Wales' - assuming William is created Prince of Wales
then they will diverge:
on William's accession George will become HRH The Duke of Cornwall and in Scotland HRH The Prince George, Duke of Rothesay. He may be created Prince of Wales by his father and so become HRH The Prince of Wales before becoming HM The King.
Charlotte will become HRH The Princess Charlotte. She may be created 'The Princess Royal' at some stage but not while Princess Anne is alive as there can only be one holder at a time.
Mountbatten-Windsor will only appear with George's grandchildren as his children will be HRHs but any children from sons who are not his eldest child will pass on Mountbatten-Windsor as a usable name.
Harry will also pass on Mountbatten-Windsor through any sons as will James, Viscount Severn if he has sons.
Louise uses Mountbatten-Windsor as she isn't using HRH Princess (and I am not going into that debate here).