I was thinking the other way around. It seems likely that Bonnie Prince Charlie would simply have been executed if they'd caught him, but if someone was stupid enough to exile him to Australia we could see a rival country fighting for control of the Empire. It would certainly be advantageous to the Americans to have the British government on the other side of the world, and they might throw in their lot with the Jacobites if Charlie's descendants grow to love Australia and don't want to leave.
Given that a failed rebellion is much more likely than a successful one, I love this idea. I'm not sure @darcykelso wants that though, since (as I mentioned) the Hannovers are the legitimate family.