WI Spain, Portugal, Brazil, and Argentina all join the war on the Axis side when Italy joins in June 1940? (or alternatively timed with Japan's attack in December 1941)
Their ability to seize European territories in the South American periphery would still be limited, but the shock value could be significant; especially if combined with early coordinated Axis attacks on Gibraltar and Malta. All of these countries will still ultimately be devastated by the economic might of the Allied war machines, but a massive third front in South America could extend the war a year or two.
Engineering truly strong fascist governments in all of these countries at precisely the right time tables to get them to join the Axis at just the right moment would be tough. It would be even harder convincing them to commit their weak economies and military forces to an essentially suicidal set of strategic objectives for the benefit of Germany with only a slim hope of actually reaping rewards from a global Axis victory. But if you could manage to get all of them to somehow join early in the war in at least a semi-coordinated manner, it would certainly give the Allies some more headaches (not to mention make for a pretty interesting TL, especially considering the aftermath and implications on an ATL cold war).
The Kriegsmarine and Luftwaffe would have significantly more bases to operate from, at least initially: Iberia, Macaronesia, and the eastern half of South America. What would be the goals of these Axis partners? Uruguay seems achievable, as would Paraguay. The Guyanas probably would not be that easy but still doable, and would almost certainly be the first targets of the US which might join the war way early once it realizes the Axis has spread to the Western Hemisphere. Argentina taking the Falklands would probably require some of that insane early war Axis luck. Maybe Argentina tries to take over parts or all of Chile to establish the very beginnings of a hypothetical connection between the Japanese Empire in the Pacific and the U-Boat infested waters of the Atlantic? A quixotic Axis drive towards the Panama Canal would be possible only with some kind of coup in Colombia. Or perhaps a suicidal Fallschirmjager commando assault could be launched to temporarily sabotage it. The Nazis would have to do their damnedest to push anti-imperialist propaganda throughout Latin America. This would probably work best before invading the USSR to help cultivate the anti-capitalist narrative. The conflict between Peru and Ecuador might provide additional exploitable chaos to the region and could possibly be integrated into the global war as well one way or another.
Ultimately though, giving the fascist elements in these four countries enough power, then giving them enough economic strength to quickly develop even remotely potent military forces and logistics, and then somehow giving them enough motivation to sacrifice the benefits of neutrality (and forfeiting distant colonies in the case of Spain and Portugal), and managing to get them to coordinate their actions in an effective way is pretty tough to put it mildly. It would certainly make for a great ATL to have a major Latin theater for the Axis to wreak havoc in at least for a while.