Expanding on my previous post, while it's an interesting scenario to ponder, here are some problems with the idea of a "British Argentina".
Let's say the British succeed in their invasion of the Rio de la Plata area in 1806-07 as you suggested.
You still have to overcome all these hurdles:
1. Large Spanish-speaking population:
In 1810, Argentina had a population of 600k people, with an additional 150k more in Uruguay. For comparison, New France only had 70k people when the British took over in 1760, and you still have tensions between French and English speakers in Canada to this day!
This means you would need a massive number of settlers to make "Argentina" majority English-speaking. Let's say Argentina becomes a British Dominion, the British push for massive Anglo immigration. You would need at least 4 million immigrants over the course of the 19th century to make a dent on the growing Spanish-speaking population, 4 million immigrants being a minimum acceptable number to achieve British majority. In real life, Spanish-speaking independent Argentina received over 1 million Irish, and around another million counting English, Welsh and Scots. So that takes care of 2 million, but you still need at least two million more at a minimum. This would erase the entire immigration that went to New Zealand, and roughly half of what went to Australia. So Britain would lose two Dominions to gain English-speaking majority in one. And this assuming all those settlers actually will want to move to what still is a majority Spanish-speaking country.
2. Large resistance to British rule:
Argentines expelled the British twice. There was a clear budding movement for independence at the time.
Would the British move additional troops from fighting the Napoleonic Wars so they could have what back then was a backwater of the Spanish Empire?
Sacrifice the chance to defeat Napoleon to have English-speaking gauchos and 5 O'clock tea in the Pampas?
3. What are the British war aims?
Why would the British want to hold Argentina? Britain at the time was moving from the notion of acquiring new colonies to the idea of extending what was called the "informal empire" through trade.
In our timeline, the British got free trade with the whole of South America without having to sacrifice a single troop, just by supporting our independence and maintaining friendly relations.
Why would they risk alienating the entirety of South America and losing free trade with the whole continent of South America just to keep Argentina?
4. Spain became a British ally in the Peninsular War
Most likely the territory would have to be returned.
5. What would the British do with it?
"Argentina" as we know it would most likely not exist at all.
The British might have limited themselves to holding Buenos Aires as a Hong Kong-style port to secure free trade. (Which would have resulted in a much poorer, resentful rump Argentina)
The British might also have limited themselves to expanding through Patagonia, while leaving boer-like Spanish-speaking Republics in the interior alone. (Which would have resulted in several more countries, not just one "Argentina")
The British might have invaded those Spanish boer-like Republics later on. (Resulting in a lot of tension between Spanish and English speakers to this day in this alternate timeline).
The British might have decided to grow cotton in Northern Argentina and Paraguay, since the climate is suitable, importing thousands if not millions of Indian plantation labor. (Which would have resulted in very fucked-up race relations and a South Africa-like Argentina with a stratified caste system, and poorer than in real life)
In short, if you want a POD with an Argentina as wealthy as Canada or Australia, it seems to me it's much easier to study the history of the real country and figure out what happened rather than just fill everything with magical Anglos. Seems particularly disingenuous when you already had a democratic, wealthy, "Honorary Dominion" British-allied Argentina in real life, from 1880 to 1943!
A British Argentina might have gone terribly wrong, or very right, but there are a lot of potential problems to be addressed for it to be a viable concept.
I've seen this trope on several websites, so don't take this as an attack on you in particular, seemed like a good chance to debunk it.