I believe we've gotten to a very interesting point in history where the ominpresent media will force monarchs to use the name they are commonly addressed by as their regnal name; I doubt Prince Charles or Prince William will easily be referred to as King George or King Edward.
Elizabeth Alexandra Mary
Charles Philip Arthur George
William Arthur Philip Louis
Beatrice Elizabeth Mary
Eugenie Victoria Helena
Birgitte Eva
Rose Victoria Brigitte Louise
It isn't a given that he would use Charles and considering how vilified he has been as Prince Charles at times it is possible that he might opt for a new beginning.
We will have to wait and see but since 1837 there have been six monarchs - Victoria, Edward VII, George V, Edward VIII, George VI and Elizabeth II and it is 50% who have reigned under their given name and 50% who haven't (Victoria, Edward VII and George VI all used a name other than their first given name).
Victoria had always gone as Victoria, though. Her parents had Alexandrina imposed on them IIRC, and they never really used it.
I have read that the then Duke and Duchess of York wanted to name Princess Margaret, Anne, because they thought Anne of Youk sounded nice. However, Edward VIII didn't like the name so she became Margaret Rose.
Margaret Rose (goes so beautifully together, the perfect Princess name.
But my fav must be Elizabeth Alexandra Mary, because it shows the Queens mother (Elizabeth), her great-grandmother (Alexandra) and her grandmother (Mary).