I'm thinking possibly for British Argentina, in the long haul, that at first it goes through that period of instability in much of the 19th century which corresponds to the OTL Argentine/Uruguayan civil wars and which is similar enough to what went on in South Africa at that time.
Then, if it manages to overcome those tensions while the territories are still British colonies/protectorates, and they federate into what I've described before as the Argentine Confederation, they could go in for a period of economic growth (driven by wheat, beef, etc.) and massive immigration in the late 19th/early 20th centuries like OTL Argentina. (Here, I'm not talking about most of the interior, which gets absorbed most likely into Chile, Bolivia, or Paraguay and/or remain one or more independent countries. I'm talking mainly just the Pampas/Littoral/Uruguay and Patagonia.) Major differences being that major sources of immigration include the British Isles as well as Spain and Italy (folks from the latter two countries gravitating mainly to the Hispanic sectors), and that the land distribution system is overall somewhat better than OTL (especially as far as the distribution of newly-settled land is concerned) and the political culture is overall somewhat more democracy-friendly than OTL (though shaped by the British-Spanish tensions). That's where Argentina's (or at least the Pampas') geographic advantages of easily-navigable rivers and fertile land really come into play. This time is also when Patagonia is finally settled and there are a lot more Anglo immigrants there than IOTL. For all the ethnolinguistic tensions that there are, at least it's not South Africa in the sense that there are mainly just these two white groups and not also a huge non-white underclass to complicate politics in the future the way that South Africa, and for that matter neighbouring countries like Bolivia, has.
Later on, thanks largely to this Argentina automatically entering the two World Wars due to being in the British Empire/Commonwealth, as well as because of the newfound geographic and economic advantages, it industrializes for real and doesn't resort to Peron-style autarky and import-substitution industrialization. There's no real elite Conservative movement like was found mainly in the OTL Argentine interior that is bothered by all the new immigration and the democratic gains, and thus there's no coup in 1930, 1943, or really any year. Thus, that Argentina is a good deal better off than OTL Argentina, with a per capita income on the level of New Zealand (though not necessarily Australia, Canada, or the US).
Of course, the last two paragraphs are a significant "if".