Look to OTL Portuguese occupation of Uruguay for clues. Like the Brits, Portugal (fairly) quickly occupied the urban center, but it took them several years to pacify the countryside. I don't think Britain ever tried to occupy the countryside. Their goal was to use Montevideo as a launch point to assault Buenos Aires, and once the BA goal was thwarted, they packed up and left, so we don't know how they would have fared. After several years of occupation by Portugal (and later Brazil), Buenos Aires acted as a launch point to stir up trouble in Uruguay, and ended up creating an independent Uruguay.
I do think that if Britain wanted a presence in the area, they should have held on to Uruguay, and probably could have. but it wasn't going to be easy. In an earlier colonial time, it would have made sense, and it probably seemed like a good idea at the time. They had no way of knowing that the situation was going to drastically change (Spain flips from being an enemy to an ally, and a massive independence movement all over south America erupting) within months to a few years.
I'm thinking if you mix OTL with TL, you have:
Britain controlling Montevideo. within months Nap invades Spain. Britain agrees to return Uruguay to Spain at the conclusion of the European conflict, but extorts trade concessions in the La Plata region. Britain gets the trade presence it wants, and is absolved of occupation costs.
With a British interest in the region, Portugal is going to dissuaded from pushing south. Britain is basically running Portugal while the court is in Brazil. That has butterflies for the early years of Brazil, who don't waste precious dollars trying to hold on to Uruguay. I'm sure Pedro I would find another way to spoil his reputation, but maybe without a war sapping his popularity and the country's coffers, he could have smoothed the way to a more stable early Brazil.