As a general observation about the central question, I'd like to note that "no ARW" by definition means "Britain doesn't settle Australia". The Thirteen Colonies were Britain's dumping ground for any and all undesirables, and only after losing those did the notion of using Australia for that purpose become a real consideration. If he ARW is averted, putting effort into distant Australia becomes something that's not worth it to Britain.
This kind of puts the whole "how big would it get" question into perspective. A gain compared to OTL
here is generally a loss compared to OTL
there.
@Fabius Maximus already pointed this out. And the idea that a teeming race of Anglo-Americans is going to spread out over the world isn't exactly likely, either. They'll be far too busy settling their own continent. (Compare russia, which didn't have a big colonial empire across the seas. why not? Because the Russian East
was its colonial empire. Just as the American West was -- and will be, in the ATL -- its "settling ground".)
That being said, Westward migration may not look the same as it did in OTL. In particular, with all of Anglo-America being part of one country/empire, I'm very doubtful that OTL's Mexican Cession would occur. It's often posited that Amerians just moving pretty much in a straight line to the West in an unstoppable tide is somehow inevitable, but let's consider that the USA in OTL was pretty radically decentralist for decades, its industrial efforts were regional, organised on the state level, and there was a lot of agriarianism that drove settlers West (to settle homesteads). Is this inevitable? Certainly not. If the ARW is avoided, and reform is effected, then I'd expect "Jeffersonian radicalism" to be rather still-born. The (pro-British in OTL!) Federalists are a closer approximation of what Anglo-American politics would look like.
Naturally, it would not be quite "pure Federalism". Specifially, Britain was moving towards free trade, and mercantillist policies of the sort that a man like Hamilton favoured just aren't going to fly anymore. Using high tarriffs to fund industrialisation isn't going to happen. But on the other hand, more centralist policies would be likely. it wouldn't be the "states' rights bonanza" of the OTL Antebellum USA. I'd expect the colonies to be united into multiple Confederations/Dominions/whatever, and I'd expect each of those to be authorised to carry out projects crossing state lines. As such, various canals and other such works would probably get completed earlier. This favours industry, and it provides
jobs. That automatically reduces Westward expansion: there's just more economic incentive to stay near the East coast, at least initially. So Westward migration may at least be delayed just a bit, and possibly may be reduced compared to OTL.
It also seems fairly credible to me that if the ARW is avoided, it will be because Britain essentially gives the colonies considerable autonomy (which is far more likely than representation in Parliament). This would include the right to settle the already-British-held West, so long as established treaties with Indian tribes are respected. (Future treaties wth tribes further West wuld be unlikely, though-- but I'd expect the forced relocation of the Five Civilised Tribes to ne a no-go in this scenario.) At the very least, this takes some initial pressure off. More than that: animosity between Britain and the USA over the North-West Territory in OTL led to delayed settlement (Britain was still maintaining a presene there, and then supported Indian resistance to US encroachment). This would not be the case here, so the angle of migration in the period 1776-1815 would be more towards the North-West than OTL, and somewhat less towards the South-West.
Britain laying claim to Louisiana at some point (nabbing it from either Spain or France, depending on who holds it at the time) is quite likely. Discouraging migration into Texas and encouraging settlement of the more Northern pats of the vast Louisiana Country could be easily done by stationing troops at New Orleans to prevent illegal movement into Texas, and similarly making troops available to protect settlers in the North from Indian raids. With this protection, settlement further North would be far more attractive than in OTL (and far more attactive than illegally trying to move into Texas, where there would be no protection, and the authorities would be hostile). So again, migration would be slanted more towards the North-West.
The borders of Lousiana Country would probably not be the exact same as they were agreed upon in 1818(?) in OTL, but would be roughly the same. Britain would presumably be interested in putting a Northern limit to Mexican claims, and this would probably be the same general line as we see in OTL. Spanish/Mexican activity never really achieved any real success North of that point anyway. So we may presume that Oregon Country would still fall into Anglo-American hands by default. There would hardly be a race to get here, though, and it wouldn't be something Britain is opposed to.
All in all, I'd expect Anglo-America to look like Canada + the USA minus the Mexican Cession. Note that even if there is at any point a Slavocrat-pushed drive to annex more Southern land, Mexico wouldn't be the obvious target. It would be the Caribbean, and given the correct circumstances, there may even be a majority in British politics to annex all of the Greater Antilles. In any event, it's far more likely that any ATL "filibustering" would be directed
there, rather than Mexico. (With all of OTL Canada wide open, and continental migration slanting more North-East, the idea of settlers pushing towards California just becomes rather unlikely.)
There is a very real chance that the increased British commitment in North America is detrimental to its OTL commitment elsewhere. If Britain nabs up Lousiana Country, does it still get to grab, say, the Cape Colony? Without the ARW, it isn't even certain if there's a French Revolution. The OTL opportunity to seize the Cape Colony may simply not present itself, and even if theoretically possible, the increased commitment in the Americas (and lack of interest in Australia), combined with American slavocrats agitating for expansion, may welllead Britain to prioritise annexing Caribbean possessions.
It is entirely conceivable that, by 1850, the British Empire in a scenario such as this one consists (outside of the Home Islands) of OTL Canada, the OTL USA minus the Mexican Cession, basically the entire Caribbean, all three Guyanas, a small South African outmost (centred on Port Elizabeth) and India. And in addition to that... well, who knows? Considering the more Westward orientation of the Empire, maybe Britain puts more weight behind its attempts to get its hands on the River Plate? That would be another location suitable for settlers, which of course reduces the number of bodies you can send elsewhere. (This would imply that OTL's ventures in Africa and the East Indies are probably not going to happen, and that the Raj may see fewer white people than in OTL, and there might be fewer efforts to push its boundaries outward.)
Naturally, that's just a guess. A possibility, not by any stretch of the imagination the certain outcome. I'm just painting a picture of what a "no ARW" British Empire
might look like on the map. My point is that it doesn't look somehow incredibly bigger than the British Empire in OTL. And my guess would be that this would hold true for most scenarios.