I think South America would take the pragmatic route and become "neutral but biased" toward Germany. The colonies would be facing a two government situation, one government representing the government in exile from the home country, and the other representing the de facto, and de jure, government at home. Vichy France would pressure it's colonies to accept German "advisors" (maybe only 50,000 or so) and the influence would have spread to adjoining countries via fifth column activities. The "Aryan Revolution" would spread as more and more peoples were made "honorary Aryans". The end result would be Europe, Africa and South America under the Nazis, with Russia as far as the Urals either under German control or being pressed hard.