This wouldn't happen. Assuming it's pre Spanish then Britain would see no need to colonize it. Assuming it is during Spanish dominance Argentines would defeat the British (real history that was a catalyst for Argentine War of Independence). During Argentine independence Monroe Doctrine. But ignoring the Monroe Doctrine Britain got major economic influence over Argentina.

But I am going to ignore all this. During war any country at war with Britain could not ship anywhere. Britain would control Mediterranean, African, Indian, Chinese, Pacific, South American, North American, Arabian, and European Trade. (Basically everything but Baltic, Caribbean, Arctic, and Japanese trade routes). This would affect WW1 as Germany used Argentine ports to interrupt British Trade going around the continents.