The question is not if we will get a new James Bond, the question is simply when. Daniel Craig’s version of the character is about as done as it could possibly be, but does that mean that all the actors who appeared alongside Craig are also finished? Ben Whishaw, who played Q in three of Daniel Craig’s Bond films certainly isn’t expecting to return, and it doesn’t sound like he’s that interested.
Daniel Craig’s James Bond tenure was unlike those that came before because it had a clear beginning and even clearer ending. And yet that doesn’t prevent characters like Q or M from being portrayed in the next Bond movie by the same actors that played them in the previous one. But Ben Whishaw tells Radio Times that he is not under contract to appear in any future Bond movies, and he hopes to see the franchise renew itself, which would seem to indicate a strong willingness, if not a desire, to pass the torch. Whishaw said…
I was only contracted to do three films, and they’re done. How many films have they made – 25 or something? I think they have to keep renewing it if they want to keep going!
There is certainly a history of actors who played supporting roles in the franchise continuing even as the lead actor changed. Most notably Desmond Llewelyn played Q starting with the second James Bond movie, From Russia With Love, and continued to play the character through the second to final Pierce Brosnan entry, The World is Not Enough. If you watch the James Bond movies in order you'll see him more than anybody else. Of course, there is at least an argument to be made that the reason for that is that despite the decades that pass: the James Bond character played by Sean Connery and Pierce Brosnan are supposed to be the same man.
But even in the case of a definitive reboot of the franchise we’ve seen characters carry over. Judi Dench played M in Pierce Brosnan’s last franchise entry, then reprised the role in Casino Royale with Daniel Craig and continued on even after the character's death in a cameo in Spectre. Dench appeared despite the fact that those two actors were clearly playing different versions of James Bond. The movie simply never addressed it, and everybody just moved on.
The same thing could potentially happen here. Whether we get a brand new origin story for James Bond, or we simply get a Bond movie that picks up in media res and doesn’t reference the fact that James Bond died in the previous film, the next James Bond will not be the same character played by Craig. But it’s unlikely that any fans would be that bothered by seeing the return of Whishaw or Ralph Finneas as M. Before we know who the next Q or M will be, we need to know what the plan is for James Bond himself.