It depends on the POD. Personally I tend to accept a form of the "frontier" thesis, that the ambition and possibility of making a new start out west transformed American colonial society into something undisciplined by British standards. It would be a question of whether Imperial authority would have goals more or less in line with those of the unruly colonists--but if we assume conflicts of interest arise as they did OTL, then keeping the American continental colonies, or even just a subset of them, under the British crown would surely involve coercive measures. If we stipulate these measures are successful and do not merely delay independence, I suspect their nature would have to transform the social situation from OTL considerably. The colonists would have to be effectively restricted from moving west in order for the Crown to keep control. I suspect it is not very well founded to suppose that practice in later British settler colonies is a guide to what they'd do in an attempt to keep the BNA Atlantic coast colonies. OTL, Canada was formed out of colonies that remained loyal when put to the test of revolutionary agitation, plus a military ruled but largely quiescent Quebec that perhaps shrewdly judged they might be no better off in an English-speaking and expansive USA than they could be in a colonial situation where the British needed them, and had already demonstrated that their rule could be tolerable, and finally refugees, clearly loyal to the crown by the very act of their expulsion or retreat from their homelands on behalf of their allegiance to the king, settled Upper Canada. In the postrevolutionary settlement period OTL, the colonies remaining had thus strongly proven their crown loyalty and this would be further reinforced by the War of 1812. So Britain could afford to devolve power to the colonies, and then to countenance their federation, while supporting their expansion westward. The USA itself served as a lightning rod, drawing off any republicans or radical anti-authoritarians to a bigger country that suited them better, leaving Canada mostly loyalist, to a reliable degree.
Thus, this same model was available to guide the pattern of development in Australia and even South Africa, though complicated there by the strong Boer resistance and by the fact that the native African majority was not going to die off and effectively go away.
If Britain hangs on--well, it matters first of all how and when. Does the question of secession never even come up? This must be either because the colonies enjoy close alignment of what they desire and British policy. Given the lack of effective representation, such alignment suggests either a very astute management of colonial affairs in London that is willing to sacrifice British interests for the sake of harmony in the colonies, or just plain dumb luck. Or that the colonials have fears that cause them to look to British power to protect them beyond what they themselves think they can manage; if France or some other European power (say a Spain that in the ATL is much stronger and proactive than it was by the late 18th century OTL--which would have massive consequences for the situation of Britain in general) held strongholds near the North American colonies that menaced them, they might set aside their objects to this or that piece of imperial decree in order to keep the defense the crown provided. But that situation answers the question of expansion--no, no expansion, or anyway limited, because the rival power stands in the way. If that power is eventually beaten and the frontier is open again, I suspect the conflicts of the OTL 1770s will rise up in the wake of that,.
There is a third possibility, besides Utopian luck leading to miraculous harmony, or the colonies being hemmed in by a strong foe--which is that the colonies and metropolis do get out of alignment, the colonies rebel--and fail, the British managing to scrape up enough force and manage it intelligently enough that the rebellion is foiled and sufficient reliable military forces back up a stern new regime that suppresses further tendencies to rebel again. This situation would lead to nothing like what evolved OTL in Canada or Australia! Instead of federating the distinct colonies into a British North America, I suspect the policy of divide and rule will suggest keeping all the existing colonial divisions and reinforcing them; 13 or more distinct entities will each have their own strong, royal/Parliamentary appointed governors, who may confer with each other for informal federation of imperial power over the colonies, but will permit no federation of the colonials themselves. The colonies would be governed paternalistically, industrial development there curtailed and allowed to expand only grudgingly, with an eye toward maintaining the home island's lead. To the west, beyond the Appalachians, Native tribes who stood with the King would be rewarded with extensive protectorate territories and relied on to deny any wildcat pioneering, within their own territories certainly and also called upon to join British regular forces in hunting down others.
Perhaps over the course of the 19th century this might change somewhat, with slow permission for new colonies to form in the interior, and with the Native tribes gradually being assimilated to British norms and acquiring common interests with the colonials. But I suspect that the sentiment for independence of the colonials won't ever die out completely--perhaps never rising to a sufficient critical mass to enable another rebellion, but remaining strong enough that the paternalist restrictions won't be relaxed. Reinforcing this is the interest of Britain to remain predominant. Despite repression, very likely the American territories will come to outweigh British population, and the potential for American industry to surpass British will be clear, so repression will remain the norm to keep them divided and dependent. Relative to OTL, 19th century Britain would be worse off, lacking the vast market for British imports and scope for investment the USA of OTL offered, but to an extent they will not know what they are missing, and to another extent--fear that the continent can indeed achieve vast economic mass, but that it would then either tear loose of British control or dominate the empire by sheer numbers. Note that OTL neither Canada, with its actual settlement zone within a few hundred miles of the US border up against the tundra, nor Australia with its area dominated by the badland Outback, could achieve more than 1/10 the US population, so the issue of the Dominions outweighing the homeland by population or production did not arise. Holding the original 13 colonies and the hinterland given by the 17th century victories over France and Spain, that potential is all too clear and despite repression would probably be realized anyway, in time. And so, the British Empire in such a case must reject the principle of democracy. This may make it actually easier to maintain rule over holdings such as India or South Africa. But it will inhibit industrial development. Perhaps then the outcome would be a bit more multipolar in Europe; Britannia may rule the waves but lack the full depth of industrial power that enables her to keep her navy both numerous and up-to-date, so that rival European powers such as France or eventually Germany can aspire to compete more effectively. Meanwhile the Empire, despite having the North American population to draw on (but selectively, remembering their dubious loyalty) may have a crisis of manpower, attempting to control India and other Asian and African holdings while also dominating the restless North Americans. Perhaps slavery is never abolished, with the general repression of democratic thought and sentiments.
I think I may have seen a TL or two that explores these possibilities but I didn't like them enough to keep reading.
