Umm, as cool as this might be, I think it will be much harder to have happen than you outline.
First, without an American Revolution, how and why does the French Revolution break out when it does?
Second, while Napoleon constantly threatened the invasion of Britain, I believe I've read that his armadas of landing craft would have had a hard time, even if the RN disappears for a short time. Then the invading army would certainly have had to contend with supply problems (unless the RN really does completely disappear). And how loyal are the British people going to be to their Napoleonic monarchs? I can see Ireland beeing freed if the French more actively support the Rebellions in 1798 and perhaps thus a similar kind of ethinc division in Scotland and Wales (maybe), but in Ol' England itself, I think the population is rather resistant for quite some time.
Third, it'd be hard to instanteoulsy transport the infrasturcture of British power to BNA: there'd be no financial system to raise money, no basis of industry. If any of the old aristocracy comes with the crown, they're not going to get along well with the colonials, to put things mildly.
Fourth, wouldn't there be some kind of attempt to retake the Home Countries?
Fifth, how does a Britain not strong enough to hold onto Britain itself manage to keep New Zealand and Australia? Isn't Napoleon trying to find himself a nice Empire?
Now, the idea itself may still be workable. Indeed, I quite like the idea: it combines my Anglophilia and Americanism with of course the possibility of eternal Frenchie/Nappy bashing.
I'd say we need a clearer understanding of how the POD comes about and there needs to be something that turns the English people away from the monarchy (an earlier Peterloo massacre?). Hence, the British revolt from the crown, Napoleon/French ruler X comes in to "aid" them and the Royals are forced to flee. The RN has the choice of loyalty to country (and rebelious republican mobs) or loyalty to crown. They chose the latter and ferry thousands of refugees to America. When the King arrives, he signs the Great Charter/Second Magna Carta/Maxima Carta (equivalent of OTL 1787 Constition but with a more parliamentary system) which outlines the new form of the British Empire. Over the years, the RN continues its refugee operations, taking millions from the continuously revolting Old World (eventually with the consent of Napoleonic governments). Also, might the flag be a bit different?