I really don't think that the British would be able to take a hold of Bolivia, given the large distances that the British would have to traverse from the Plate, over very poor roads and at least potentially enemy territory. Besides which, for the longest time during the Spanish-American Wars or Independence, what is now Bolivia was a stronghold of the Spanish royalists in real life.Ahh, okay, those population figures do seem to make such a scenario more feasible.
I think your suggestion of multiple Spanish-speaking republics based around the major settlements does seem quite realistic. (what is now) Bolivia and Paraguay were part of the Viceroyalty as well, so it is possible that the British could annex those, as well.
My understanding is (often different sources give conflicting information) that the inland population of OTL Argentina in the early 1800s was concentrated around Salta, Tucuman, Cordoba and Mendoza (i.e. the area to the North West of BA). Somewhat similar to today. But Patagonia (obviously) and the area along the Parana and Paraguay rivers did not have large Hispanic populations.
Is this correct?
If so, it seems likely that the British would extend their control North of Montevideo and annex Mesopotamia, Chaco and Paraguay itself. These areas would subsequently become majority anglophone.
As for Paraguay, I don't see so much of a better chance for the British to take a hold of especially if the British don't directly control Buenos Aires, downstream, for very long after 1807. Probably, as in real life, the Paraguayans declare independence from Spain and from Buenos Aires, not interested in obeying orders from Buenos Aires. And Paraguay, so far as I know, had a substantial Hispanic/Guarani population back in those days, just about as many people as Buenos Aires province or any of the interior areas in the north/west. Much more, for sure, than in the nearby Chaco or in Patagonia.
Chances are that the British would have, on the whole, preferred British over non-British immigrants to the River Plate and so forth, just like with Canada, Australia, etc. But non-British immigrants, especially of northern European stock, did arrive in those latter countries through the early 20th centuries. There were plenty of German, and some Dutch and Scandinavian, immigrants to those countries, and also limited numbers of Italian immigrants (nothing like to the US or OTL South America, though). So, for TL British South America in the late 19th-early 20th centuries, I see many German and other continental northern European immigrants arriving, and some Italian and even perhaps Spanish immigrants arriving (though nothing, of course, like in OTL Argentina/Uruguay). Large numbers of Italians arrive in the TL River Plate, only after World War II, as was the case in Canada, Australia, etc. In an analogy to Quebec and Canada, I see the Italian immigrants assimilating, for the most part, into the anglophone sector rather than the hispanophone sector, though in Spanish-speaking areas, some Italians could assimilate into the hispanophone sector.Especially if the Hispanic population of OTL Argentina was divided into several republics, then it seems like the British would have a strong motive to annex them. They wouldn't want 2 small colonies split by a republic / republics. Especially towards the end of the 19th century when major powers were starting to rival Britain.
Assuming the British do gain control of Buenos Aires and Cordoba, and you're left with a Canada-like confederation, with English, Spanish and English/Spanish speaking regions. My big question is this: do you get a massive influx of non-British Isles European settlers?
Huge numbers of Italian (as well as German) migrants arrived in OTL Argentina around the turn of the 20th Century. This is important because I'm assuming most would assimilate to the (dominant) Anglophone culture, rather than the Hispanophone one, even in Hispanophone-majority areas. Many people assume than Romance Catholics would obviously assimilate with a similar culture (i.e. Spanish), rather than the Anglo Protestant one, but the situation in Quebec shows that this is not necessarily the case.
But if Argentina was a British dominion, would this wave of immigration still occur? Canada and Australia did not receive large numbers of Italian immigrants until after WW2, and Germans are a similar story.
Italians migrated to Brazil, Argentina and the USA so I don't think it is a question of cultural/religious preference. Canada/Australia were very wealthy and of a similar population to Argentina in 1900. Canada was easier to get to.
Was there some sort of British policy discouraging non-British/Irish migration to British dominions? Or were Italians/Germans reluctant to live in a colony of Britain?
Or was Argentina more preferable to Italians for some other reason? (maybe climate?) In this case I think we assume than Italian migration occurs as in OTL.
The reason why Italian immigrants preferred the US at the turn of the 20th century was availability of jobs, along with the desire of immigrants in general to settle in America, while the preference for South America (especially southern Brazil and OTL Argentina/Uruguay) was due to the similarity in language and culture along with job availability.
In the Cuyo, the Mendoza area is useful for vineyards, while in the Northwest, Tucuman is the sugarcane centre, but these crops became useful in those areas only because of high government tariffs against competition from more lucrative parts of the world. Otherwise, these crops would have easily lost out in Mendoza and in Tucuman. As for Cordoba province, its eastern half is a part of the Pampas, with its capital (also called Cordoba) being at the edge; west of Cordoba city, it's already the start of the Northwest/Cuyo area. So the British, if they're successful at (re)annexing Buenos Aires towards the end of the 19th century, may expand into at least parts of eastern Cordoba province; can't say for 100% sure.One last thing I want to consider: You suggested than the British might annex Buenos Aires province, but not Cordoba, so as to create a confederation of Anglo-Patagonia, Hispano-Buenos Aires, and Anglo-Uruguay. This would exclude the Western Regions.
In terms of 19th century economic and strategic value, are the Western regions of any worth? Patagonia is useful strategically and for Sheep, Pampas for Wheat/Corn, and Mesopotamia possibly for Cotton. Chaco, I'm not sure, but sparsely populated so no harm in annexing it. But for Cuyo and the Northwest, they're very arid and have no strategic value without access to the ocean. So is there any point attempting to annex those regions?