Interesting topic.
Let's for the sake of argument say that William for whatever reason refuse to become king. I can hardly imagine the courtiers will drag him screaming and kicking to the coronation. Even though I can well imagine he will subjected to some heavy pressure. Okay, Harry steps in and become king.
William will become King the moment King Charles (or, if Charles predeceases his mother, Queen Elizabeth) dies. If he believes Kingship isn't for him, he can abdicate. The abdication will need to be reinforced by all Commonwealth Realms; otherwise, it will have no legal effect.
If he makes up his mind to refuse the Throne
before the previous Monarch's death, he can renounce his place in the Line of Succession; the renunciation will not, however, have legal effect until it is ratified by the Parliaments of all Commonwealth Realms.
What about William in this hypothetical situation? Will he remain a royal? Or will the public (and politicians as well as the rest of the BRF) say: "Hey, Hey, you can't cause all that trouble and still remain a royal with all that entails."? Will he even have a future role anywhere, say becoming governor general somewhere? Or would he simply be demoted to noble and told to preferably stay in his shire?
Unless an Act of Parliament (again, of all Commonwealth Realms) is passed stripping William of all his royal titles, he will continue enjoying all the titles he had by birthright.
- If he abdicates (after becoming a Monarch) and the previous Monarch was Prince Charles, he will be His Royal Highness The Prince William. He will not be the Duke of Cambridge since upon his accession to the Throne the title would merge with the Crown.
- If he abdicates (after becoming a Monarch) and the previous Monarch was Prince Charles, he will be His Royal Highness Prince William. He will not be the Duke of Cambridge since upon his accession to the Throne the title would merge with the Crown. The difference between this scenario and the previous one is basically
THE - only the children (and not grandchildren) of the Sovereign are called
The Prince/Princess Name.
- If he renounces all his succession rights (reinforced by Acts of Parliaments), he will retain the titles he had from birth - His Royal Highness Prince William of Wales - but might lose his peerage title.
Either way, he will remain a royal. Even Edward VIII didn't lose any of the styles and titles that were his by birthright following the abdication. Obviously, all his titles as Heir Apparent to the Throne (such as Duke of Cornwall, Prince of Wales) were no longer his, and his peerage title merged with the Crown; thus, he was The Prince Edward from the moment of his abdication and until he was created Duke of Windsor by the new King.
As for an official role - I doubt it. Certainly not that of a Governor General of any of the realms.
Then there would another problem. In the present future there will be king William with his brother Harry as his wingman, so to speak. A very important role. But in the alternative future with William out of the picture, who would step in as "Harry's wingman"?
Harry's "wingman" will be next in the line of succession. If Harry isn't married by the point, the next senior royal would be Prince Andrew, followed by Princess Beatrice. In this hypothetical scenario, the role of Princesses Beatrice and Eugenie would grow immensely since only one person - Harry - would be between them and the Throne. They would likely fulfil far more official engagements on behalf of their cousin.
How much would it rock the boat in regards to the monarchy in Britain but also in regards to William's image, if he should step aside because he really, honestly, heartily don't want to be king? Wouldn't the public sympathy be somewhat lukewarm? I can imagine a king Harry would get even more sympathy in return.
Edward VIII was the nation's darling, but his abdication was viewed as a cowardly act, and abandonment of duty in times the country needed its monarch most. I believe William's image would suffer similar fate; in essence, he would be a traitor of his duty, country and people.
British Monarchs don't abdicate; they serve till the end.