Everybody seems to be acting as though British Argentina cuts off at the 33rd Parallel South and that everything north of there goes to Brazil or something. I can assure you, that is most unlikely to happen in a British Argentina TL.
Have a look at the map of Argentina. Notice the territory between the Parana and the Uruguay River, both of which are navigable. This territory has very rich soil. It is good for growing tobacco and cotton at a time when the UK needs both.
And between the Winter Monsoons and the Southeasterly Trade Winds, India is only three to five months away from Argentina with the biggest hold-up possibly coming from contrary winds at the Cape of Good Hope. And from the 1820s on, steamboats are coming into widespread use. And there's no earthly reason why a Great Britain that controls Argentina will leave Paraguay alone and not conquer Asuncion. Or maybe even go all the way to Potosi, Bolivia and Salta and Tucuman . It all goes hand in hand.
So any British Argentina is likely to be very different, all right. It's likely to have an East Indian underclass working the fields, at least outside of Patagonia, which may well be settled by convicts, just like Australia.

In other words, Australia would be a distinctly "curry flavored" Anglo-Indian as much as Southern European Anglo-colony. Everything non-white that Australia succeeds in excluding, Argentina gets in--because Indians get brought in from the start and because Argentina-Paraguay is so very rich in cash crops that take a lot of labor to grow---and the slave trade just became illegal.
So look to Argentina to become a Christian-Hindu-Muslim-Sikh religious stew as well as what it becomes racially and ethnically, and one that leaks into interior Brazil as well as Peru and Chile if those places even remain Spanish. Whether the more conventionally White Patagonia even stays a part of this Argentina becomes an interesting question.