We have to be clear if we are talking about a scenario where the revolutionary crisis of the 1770s does boil over but the rebels are defeated with Crown victory from Key West to the upper reaches of Maine, or instead a happy scenario where the specter of nationalist secessionist rebellion is never raised in the first place and British possession goes on unchallenged.
The OP seems to demand the latter but I feel that is highly improbable. The BNA Anglo-American colonists raised a bunch of protests to the British imperial system that followed a path of least resistance--assuming no political protests! In short to assume peaceful British possession going unchallenged from founding days to modern times is unreasonable, since either it assumes an enlightened success of the British constitution to take the interest of overseas settlers into due account, enough to pose some checks, costs and compromises on powerful interests in Britain, so that in some fashion or other the Colonies remain content in the evolving British system. OR to assume that British colonials lose their political souls and submit without effective protest to rule in the most probable evolving interests of British powes that be.
The third path to ongoing unchallenged British Empire in north America I think is more probable than either of these by far--and yet not terribly probable in itself, meaning I think the OP borders ASB levels of improbability--is that the crisis does come to a head, but the OTL "Patriots" are labeled (save in defeated dissident circles) as Rebels plain and simple, because the independence movement collapses and is defeated; the majority of OTL Patriots are indeed hanged separately as Franklin warned, and Britain takes measures to impose rule more permanently and effectively.
In my opinion, which I have expressed elsewhere at length, the short term form of restored rule will be harsh, both antidemocratic and anti-autonomy. For practical reasons much power and autonomy will in fact be local, but the system will rely on much divide and rule measures to check the revival of republican secessionism. I would foresee the balkanization of the continent into small bailiwicks each with a favored and structurally pro-British local aristocracy, all subordinated to a highly centralized continental command of Imperial/Royal military and other executive resources, control of overwhelming Imperial power leveraging the loyalty of Royalist native auxiliaries. Imperial power answers only to British authority, and in early days will frequently dictate to the local governments, and in early days these will generally be the opposite of democratic. Over time I expect democratic principles to assert themselves, but sporadically and inconsistently; eventually if the global Empire is to endure, they will prevail and insist on certain forms of consistent democratic representation, but perhaps not even then directly!
But this is the dilemma; the more insistent Britain is on holding the empire together and subordinating it to British central rule, the more inexorably will real power shift over time to America. If not on a democratic basis, then eventually on a basis of wealth despite structural measures taken to bias wealth toward the old capital. At some point, assuming the imperials make decisions designed to both maintain a central bond of the colonized territories to Britain and at the same time serve the interests of the already rich in seeking new opportunities for enrichment, I think either the North American territories are driven into violent revolution and hostile separation--or unity is maintained, but the center of mass of power shifts westward to America.
I was considering for instance a system wherein the Westminister Parliament, elected only by UK constituencies as per OTL, claims supreme power--but over time a system of proxies which eventually evolve to be proportional to population and are assigned by various colony governments to MPs of their choice to represent them develops, with specific measures affecting the Empire or particular sectors of it activating these proxy votes. If they were proportional to the UK's current 600 MPs, counting all of Ireland (if the Empire will not let north America go, surely not Ireland either) for a population of 71 million, then the 370 millions living OTL in various North American regimes, all but the USA part of the Commonwealth, would among them have 3137 proxy votes! These are cast by British MPs, but handpicked as trustees of various American interests. Insofar as development converges standards of living to produce similar cultures, social stratification and thus dispersals of interests, if the proxies each ATL colony has are distributed more or less proportionally to various parties freely elected by the colonial populations, then the various parties should correspond to at least some of the British parties in Parliament, and thus MPs with suitably close interests can be found to vest them in. If at the same time as the American proxies are issued, the colonies of Australia and southern Africa are given some in proportion to OTL populations today, this combined southern hemisphere colony vote would amount to 900, with South Africa being the largest (with all of its people voting of course) with 450 proxies. Thus, unless nations such as India were counted on a one person, one vote equal to a Briton's for Parliament basis, a system like this could have worked for OTL "white" settler colonies and even rope in millions of subject peoples and give them full votes too, but with Britain still the weightiest member--but include the territory comprising the modern USA and assume it is as populated and developed, and it will vastly outweigh the rest of the Commonwealth and if allowed, rule the whole thing. My theory is that if Westminster leadership is watching carefully to prevent that from happening, and yet they do not wish to alienate their American colonies wholesale, their resolve to keep the Americans from running the show will gradually weaken and be compromised away well before the 20th century ends and we'll wind up with a Commonwealth Empire that on paper is ruled by the British metropolis, but is really subservient to American interests.
The OP of course assumes another path, that 1) apparently there never is a rupture of relations between American colonials and their British rulers, which I say can happen only if American colonial interests are not too frustrated and therefore the American settlements expand and become the massive tail wagging the British dog as I suggest they would above even if the colonies were retained by harsh means and run for generations by masters keen to keep the thumb well on colonial aspirations. And 2) are let go peacefully someday. Well, perhaps if Britons can see that the North American colonies must wind up running things someday, they do cut us loose, voluntarily, and hope that their control over Africa, Asia and Australia will never be so undermined. But I think that is a lot more vision than we can credit politicians with generally; the ones who foresee American dominance will take measures to keep Americans down before they start thinking of exist strategies. If they do that, the separation will be delayed, and will be hostile!
The wisest thing is rather to reason that the American colonies are in the image of Britain, are a larger Britain in fact, and should be allowed to rule. In that case, there is no separation ever.