In April, Liz proposed Chuckles succeed her and the Commonwealth approved. He can choose any name he wishes. The ding bat could choose King Mo.
The Queen proposed that Charles succeed her as Head of the Commonwealth - not as King. While the Queen is both (and she is the first Head of the Commonwealth), there's no actual law or anything else that states any official succession for the position of Head of the Commonwealth - theoretically it could have gone to somebody else (and some people were actually suggesting some elder statesman from a Commonwealth country might be given the role).
Theoretically the Prince of Wales could choose any name on his accession to the throne but it would be extraordinary for him to choose anything other than one of his four names - Charles Philip Arthur George - and for various reasons, only Charles or George are likely among those choices.