ATL South Americas

England isn't enough of a major power seafaring wise to be able to annex and defend all of South America. The Spanish are still the #1 sea power and Portugal a close #2 in the 1500s, and attempts to take the colonies post-1600s can only end badly for England.
 
England isn't enough of a major power seafaring wise to be able to annex and defend all of South America. The Spanish are still the #1 sea power and Portugal a close #2 in the 1500s, and attempts to take the colonies post-1600s can only end badly for England.

How about the latter half of the 18th C, or the early 19th C (after 1815) when England was gaining matery of the seas, and Spain and Portugal were spent forces as colonial powers?
 
You could have an Iberian union surviving (a scenario where Portugal never splits from Hapsburg Spain) and dominating all of Latin America, due to increased strength this could include New Orleans, Florida and everything in between that and New Spain.
That is as close as you are going to get to either Spain or Portugal dominating all of South America.

For the British it is very much impossible as they are too late into the race. However later on in history as most Spanish possessions gain independence Britain could have a little more interest and take a much more influential stance and turn most or all of the new nations into British protectorates (as it very much did with Argentina in OTL) and thus be "in control" of South America.
 
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