Of course it would depend on the details and then there would be an endless multiple of ways it goes. Thoughts as follows:
a) Without the revolution means that never takes off, rather than it being defeated. The latter is a different option and may be better because it's more likely to prompt reform in N America and probably earlier reform in Britain as well. However without it occurring depends on circumstances.
i) Does Britain accept that the colonies won't contribute to their own defence other than militia units in wartime? Or find some way acceptable to enough of the population that the US will pay for part of their defence costs in peacetime? The 1st, although it would seem unjust isn't impossible because Britain as well as the colonies got a lot out of mutual trade, although with taxes levels in Britain being so high, in part because of debts for war in defence of the colonies, its likely to be an increasing sore in Britain, especially if democratic reform isn't delayed there.
ii) Does Britain risk more conflict with the Indians by reneging on agreements to protect their lands from seizure or possibly never make such agreements in the 1st place. This would reduce tension with the colonies but probably mean some bloody wars. The Indians might fare a bit better than OTL but probably not much, even those seeking to fully integrate such as some of the southern tribes. Partly there was the desire for land without regards to how they got it plus the invention of the Cotton Gin, which might be slightly different but is still likely to occur, will make slavery and large plantations economically very profitable. As such that in itself will dominate huge areas in the deep south and hence increase land hunger.
iii) What happens with Canada? Assuming Britain still offers guarantees about their society and religion their likely to remain loyal but without the ARW and flow of loyalists north the population could remains very low and overwhelmingly French speaking and Catholic with a pretty conservative society. Probably what's now southern Ontario is likely to see settlement as OTL did before 1812 and assuming no later rebellion in the area could be settled even earlier and with larger numbers.
b) Is there a later rebellion, whether or not it succeeds? Again depends in the circumstances but assuming, with one exception no? [Largely for simplicity sake here] That exception would be if the empire moves against 1st the slave trade and then later slavery itself. In this case you could see some equivalent of the OTL USCW with probably at least some of the Caribbean colonies included. Although in those the very small proportion of whites in them would make such rebels very vulnerable. This is likely to occur earlier than OTL because while the combined might of Caribbean and southern colonies slave interests might delay emancipation I think it will still come eventually, probably by the late 30's early 40's. [Could delay it in two ways, their political power, especially if Parliament reform was delayed, leaving more room for corruption and bribary by wealthy interests and also because OTL Britain compensated slave owners. That's obviously going to be a hell of a lot more expensive if including say a couple of million slaves in the southern colonies.].
I would expect such a rebellion to be crushed fairly quickly however. Britain has both the almost certain support of the northern colones and also three things the OTL union doesn't have, i.e. a sizeable regular military force to deploy, overwhelming control of the seas and hence also coastal waters and the most powerful and developed economy in the world. Quite possibly more than OTL given the extra scope for investment in BNA without trade barriers. Plus a Parliament in London committed to ending slavery is probably going to be quicker introducing something like the emancipation proclamation. Especially since some figures at least sees this as a way both of weakening the rebels and avoiding/reducing the cost for any emancipation as the government - or whoever was doing it - would only have to pay for freedom for slaves owned by loyalist settlers.
c) Does Louisiana get added to the empire? - I would say almost certainly. There's very likely to be another war with the Bourbon powers, whether as conservative monarchs or as a revolutionary France with Spain being reduced to a satellite. The only exception would be if butterflies meant Spain avoided that role and stayed opposed to any revolutionary France. If Britain fights them as monarchies, which is probably likely to occur soon without the ARW draining treasuries and providing an incentive to avoid war, then New Orleans is too critical to development of the interior, much of which is already in British hands and too vulnerable. Also especially if Britain has protected the Indians to some degree there will be desire for land for settlement, especially if the cotton gin has been developed. Probably already clashes with settlers crossing the Mississippi further north. Simply the demographic gulf between the two sets of colonies is too great. Especially since here Britain and the colonies are a fairly united bloc.
California, Texas and other areas are less likely but still very much so for similar reasons. Unless Spain or possibly a Mexican successor greatly improves both its government stability and the development of its northern lands their going to be too vulnerable. Ditto with areas further north as with colonial demographics and British naval and industrial power I can't see Oregon or even eventually Alaska staying outside 'British' hands. In the 1770's there was tension with Spain both over the Oregon region and also the Falklands and either might be the trigger to a war that for general world purposes replaces the ARW as the nest battle between Britain and the Bourbons. [There is still the chance if British diplomacy is bad and we get isolated that this could go badly for Britain but without the colonial revolt its still likely to end up as a British victory. Especially with say a deal between Britain and the colonies whereas in return for general protection of Indian tribes east of the Mississippi, Britain supports colonial settlement west of it.
Alaska is likely to be more difficult but without a US you could see it being lost in some Crimean War equivalent or simply as OTL the Russians need the money and find it too expensive to maintain, even if the only practical buyer is a Britain they might not be on the best of terms with.
d) Does the French revolution still occur? Probably as France was already in debt before the ARW, more importantly was grossly incompetent in terms of handling its debts and also as mentioned above there's still likely sometime in the 1770's or 80's another war with Britain that's likely to increase those debts further and also be seen as another defeat, undermining the regime. Of course there's always the possibility that given more determined action the revolution might be suppressed in its early stages, although without serious reform, which seems unlikely that could simply mean a somewhat later and more explosive uprising.
If the revolution occurs its still likely to prompt intervention by the more reactionary continental powers, especially Austria since Marie Antoinette was after all an Austrian princess. Britain was more reluctant to intervene initially and may be more so if the butterflies from no ARW means reform in Britain is more successful. However assuming the regime gets more extreme, including execution of the monarchy and other leading figures and/or it starts being very successful in terms of moblishing mass armies and starting to overrun much of western Europe Britain is likely to become a determined opponent. Its possible that it might, like the Americans OTL stay fairly moderate in which case you could even see Anglo-French relations being good for a while although since its likely it would mean a markedly stronger France in the medium and longer term. However given the probably concerns about reactionary elements, both external and an unpurged clergy and aristocracy I suspect this is unlikely.
Assuming that the revolution become fairly extreme and engages in major conflict for an extended period with its neighbours its likely that at some point a military strongman is going to emerge. Establishing a republic in a state with a long history of autocratic rule, with minimal literacy and education and under constant military pressure, even if its often successful in its land wars is going to be difficult so I would suspect increasing disorder and corruption as OTL and a military figure to take over. This need not and quite probably wouldn't be Napoleon and whoever it is might stay a dictator in a formal republican 'government' rather than making himself a king/emperor. Still likely however with the personality culture that often occurs you get something like OTL Napoleonic wars although possible a wiser leader manages to avoid continued conflict until their defeated and possibly establishing ultimately either a new French dynasty or a 'republic' of some version that could end up be dysfunctional with succession problems.
e) Do you get a unified NA state? Probably not I suspect, at least in the 1st century or two. Already before the revolution there was talk of establishing new western colonies and this is likely to occur OTL while there would be strong cultural differences. Without a clear external threat there's no great incentive for a unified central government and probably a lot of colonies would prefer to keep it that way. Also until railways provide a greater sense of unity and ease of communication the state would simply been too large and diverse for easy cooperation. Plus as some said in the AH thread there is an incentive for Britain itself to encourage regional assembles rather than a central one, to prevent a single such bloc dominating the empire as a whole. If such groups are established and develop their own cultures and economic systems then they could also develop their own identities and decide submersion into one massive state is not in their interests. I think its only likely when either there's a clear external threat, a real tension with Britain or a desire to spread 'American' power outside N America, possibly establishing their own colonies and overseas interests. Of course some of those factors could be combined to some degree.
f) Getting back to the original question of what such a British empire would look like then I would think it would include Britain, most/all of N America, at least north of the Rio Grande sort of area, much of the Caribbean, India -as the basis of the EIC control was already starting and its too rich to be allowed to come under someone else's control. I would suspect that Australia is likely to be British simply because I think British control of the seas is still likely and hence its easier for Britain to construct a colony there than anywhere else. Especially if there's a prolonged event like OTL Napoleonic Wars which retards the continents industrial development and restricts their access to the rest of the world. Cape Town and a few other such places are likely as key staging posts for deployment of naval and economic power. Ditto when either a canal is built at Suez or Egypt comes outside Ottoman control its location on the 'short cut' to India is going to make it something Britain will want to have at the very least outside of hostile domination. There will be colonies in Africa simply because your still likely to have attempts both to stamp out slavery and also to introduce 'western civilization' aka Christianity to the region. Plus when other powers start to catch up in industrial development you could see some sort of scramble for Africa. Depending on circumstances in both Britain, N America and the rest of the world, especially Europe as to whether its markedly larger or smaller than OTL. I would expect the Spanish empire to collapse at some point into independent states probably with at least implicit British support but that it would be mainly seen as a trading area and zone of influence as OTL rather than any direct colonisation of the region. At least outside possibly Panama or other equivalent for a canal or the American colonies themselves seeking to expand further south.
Anyway waffled on too long but initial thoughts.
Steve
