The satellites are not stationary, but go around the earth. This means that to get good coverage in the US, they actually have to provide good coverage everywhere near the same latitudes (on both sides of the equator). Since the satellites in the southern hemisphere will have nothing better to do and have less demand, they are probably going to start rolling out service everywhere they can at the same time.
Most of Brazil is actually a bit out of luck in that it's too near the equator, so satellite density overhead will not be that great anywhere but the very southernmost areas until there are lot of the birds in the sky, so you won't have service immediately. But places like Chile and Argentina will have great access pretty quickly.
However, as a caveat to that the receiving antennas will be expensive at the start and SpaceX still has to negotiate access to spectrum in every country separately, and in poorer countries this will most likely have to be paid for, so that will hike the prices up.
