Okay, so before I starting speculating about how this BNA is governed, let's consider the wider picture of BNA.
Now, there's something of a problem, IMO, with the POD since we have to wonder what France does for all this time, a period when French power was at something of a zenith (Louis XIV) and English/British something of a nadir (British Civil War). I could easily see the French try to focus more on LA, which would be cool in its own right, but for the sake of the TL let's say that France gets involved in European conquest, being at some point in time marginally more successful than it was OTL.
The next pre-requisite is that we figure out what happens to New Amsterdam. I don't have a problem assuming the English take it, but that was in the context of the Anglo-Dutch wars which means that we still need to know what's going on in Europe. I could easily go with some sort of scenario that sees the Dutch threaten by France and perhaps conquered leaving New Amsterdam almost by default to the English/British.
Next, consider the state of affairs as it appears to the native residents of the New World: while in OTL they could take some comfort from their ability to play the British, French and Spanish off one another, in TTL they have no hope of doing so. This will make them much more likely to resist colonial encroachment. Alternatively, though, because of the likelihood of more continuous conflict, the colonials may change their attitude. Here, contests with natives are not likely to be proxies for European struggles. Attempting to be more upright in dealing with them may prove fruitful, particularly as Britain is not likely to offer much help in dealing with the Indians in the first place.
And this brings us to another important consideration: Britain has little to no need to station troops in the colonies to defend them from other colonial powers. This means that if the colonists get taxed at all, they are liable to resent it. Furthermore, the colonists will not have the expereince of working together during conflicts with European powers to gain a sense of a wider identity: without the Seven Years' War and the need to organize aganst the French, there's no Albany Conference. However, it does seem likely to me that European powers may try to descend on the American continent during any war with Britain, which may mean the some kind of naval protection is more necessary than it was OTL. This could provide a source for huge unity as American colonial sailors mix with British ones. Perhaps the attitudes of the Royal Navy will prove different than the interactions of OTL's British Army and Colonial militia, if only because TTL's naval service is more likely to be integrated than OTL's army/militia.
It seems to me that if the colonists do form any cooperation plans, they're going to be regional. Indeed, the Albany Plan of Union is rather incorrectly remembered as a truly continental plan because no colonies south of MD attended. Indeed, the reason they didn't is because many were at a conference in Viriginia spnsored by Gov. Dinwiddie. The reason for the divergence is simple: the southern colonies dealt with different Indian tribes than the northern ones did. Massachussetts had no interest in settling squabbles between the Carolinas and Catawbas just as Virginia had very little interest in the dealings between New York and the Iroquois/Haudenosaunee. And then of course there's the huge distances involved and little intra-colonial comunication (OTL postal connections weren't coordinated until the 1750s, when Franklin was appointed Postmaster). I wont say too much more, because I go off on the topic of regional unions everytime we talk about colonial America.
Suffice to say that TTL's British North Americans will not lack reasons to revolt, since they probably won't get too much attention from Britain (or that might help, since nothing was better for the colonial relationship than Britain not trying to run them, at least until things got so bad the colonists begged for it). However, I see a lot which may make them less likely to band together and more likely to fragment into smaller groupings if they ever acheive indepedence.
Nevertheless, the first question in my mind is the need to figure out just what Britain is up to through this 17th century. A continued Stuart dynasty would be far different from Cromwell or William & Mary. Prince Henry surviving would be a far different ruler than his brother Charles. And then there's the issue of ATL children to throw things for a loop, to say nothing of wars in Europe.