At the time of the Monroe Doctrine in 1823, Argentina claimed the islands but no one was living there. A few years later Argentina sent a garrison which was ousted by the British in 1833. It has been British since then except for the two months of Argentine occupation in 1982.
I don't expect the Latin American countries would side with the UK or the US on this. True, the people living there now want to be British, but the same is true of the majority of the people living in the parts of the US that briefly belonged to Mexico. I imagine that if a poll were taken of public opinion in Latin America and they were asked "should the US return to Mexico the lands it acquired by the Treaty of Guadalupe Hidalgo and the Gadsden Purchase," most people would say, "Si."