both countries were engaged in a very expensive arms race in the early 1900s, so war was definetely possible, but I can't think of any good PODs that would actually trigger it. As far as I'm aware there were no territorial disputes between Argentina and Brasil, and they never had any border skirmishes or anything of the sort. Also Britain and the USA had strong commercial interests in the region and they would probably intervene very quickly to mediate a solution.
Regarding Uruguay, they would be under a lot of pressure. The Argentina-Brasilian border has very poor infrastructure, if you look at a map of latin american railways you'll see that there was practically no way to support large scale troop movements. There are some very dense jungles in the area with lots of mountains and hills, and in the summer temperatures can go as high as 40 C (104 F) any armies trying to make it across will suffer heavy attrition. In fact, in the Argentina-Brasilian war of the 1820s, nearly all the fighting took place in whats now Uruguay. Both countries would seek access to Uruguayan territory, and Anglo-American guarantees would probably stop them.
As for how the war would actually go, early in the 1900s Argentina had a clear advantage, during this time the Brasilian navy was underfunded and suffered a series of mutinies. However Argentina's position was very vulnerable, all their exports went through just a couple of small ports (Buenos Aires and Rosario) which means that a blockade of the river Plate could have completely destroyed their economy. Brasil's best chance is to attack before the USA delivers Argentina's new dreadnoughts, sometime between 1910 and 1914. Depending on how the conflict plays out there is a small chance of Chile joining (just a few decades before, Chile had almost gone to war with Argentina over Patagonia, and tensions remained high) which could also drag in Peru and Bolivia. But my best guess is that the war would only last a few months since American diplomats would pressure everyone into signing a ceasefire. During this time neither side would be able to gain any ground and thousands of soldiers would die of disease and exhaustion.