Lady Louise Mountbatten-Windsor is the only one of Prince Edward and Sophie Rhys-Jones's children to retain the surname Mountbatten-Windsor. Her brother, James, Viscount Severn, is styled as James Mountbatten-Windsor, but he will not use the surname in his adult life.
The reason for this is that when Prince Edward and Sophie married in 1999, it was agreed that their children would not be styled as prince or princess. This meant that they would not be entitled to use the surname Windsor, which is reserved for the direct descendants of the monarch.
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However, the Queen granted permission for Lady Louise to use the surname Mountbatten-Windsor, in recognition of her grandfather, Prince Philip, who was born Prince Philip of Greece and Denmark and took the surname Mountbatten when he became a naturalized British citizen in 1947.
Prince Harry and Meghan Markle's children, Archie and Lilibet, do not use the surname Mountbatten-Windsor. This is because they have been raised outside of the British royal family and do not have the same status as Lady Louise.
It is possible that Archie and Lilibet may choose to use the surname Mountbatten-Windsor in the future, but it is not clear if they will do so.