My question is what happens after the British take the colony by force. Could we expect a US-style revolution?
Maybe, but it would simply go the way of Lower Canada's Patriote rebellions of 1837 - there, numerous French-Canadians (and some anglos) fought against British authority, intending to create a republic in Quebec but without success. A similar rebellion occurred in Upper Canada at the same time, and Louis Riel and the metis fomented one rebellion in 1870 and another in 1885 (both in the Prairie provinces in Canada) - all were put down by the British or Canadian government.
As for impacts of a successful British conquest on the rest of South America, if all of OTL Argentina and Uruguay is under British control, that would mean more British influence on the continent than just economic, and the surrounding countries (i.e. Peru, Brazil, Chile, Bolivia, and Paraguay) are all somewhat wealthier per capita than in OTL.
No, they would have a choice. Unless I'm mistaken there is not a large, far more fertile and populated British colony bordering the Viceroyalty of La Plata.
When the British took over the Dutch Cape Colony in present-day South Africa starting in 1806, the Afrikaners didn't resist British rule, at least not to the point of actively fighting for a return of control of their colony, except for the Great Trek 20-30 years later and forming their own inland republics. The point is, that the Cape Colony, like a would-have-been British Rio de la Plata colony but unlike New France, never bordered a large and populated British colony or series of colonies. So in the case of the Viceroyalty, the hard-core criollos would have maybe fled Buenos Aires for Córdoba and other inland locations (as the Marquis de Sobremonte did), and they would have fled to neighbouring countries like Chile once the inland locations would have been captured.
Not to mention that the conquest of Buenos Aires, or all the Rio de la Plata settlements for that matter, would not automatically give the British control of all of OTL Argentina as these threads always assume. The interior would continue to be ruled out of Córdoba.
True, the interior would still be ruled from Córdoba; however, the British would have pressed on to Córdoba and to other interior locations in due course once they were finished with conquering the Rio de la Plata. Actually, the British had plans to capture Valparaiso, Chile, once they finished with Buenos Aires and Montevideo, but to consolidate the gains made in the Rio de la Plata, it would pay more to go into the interior. A lot closer, and you wouldn't have to cross the Andes or sail around the treacherous Cape Horn like you would going to Chile. Besides which, Buenos Aires did control the entire Viceroyalty of La Plata as well as its own intendency, and the British would have been intent on finishing off the entire Viceroyalty. So Córdoba would have been the next logical major step in taking over that part of Spanish South America.