What I am saying is you have no idea what the consequences of thousands of nuclear devices off at the same time are. Too much warhammer.
Nuh uh warhammerz isn't a valid argument.
Radiation isn't magic. It won't kill all humans instantly everywhere.
The nuclear winter, while debated, is the more likely culprit, but even the climate doomers don't consider it a human exterminator, as ice ages had happened before.
So no, it wouldn't be the end of the world. Some scources even say that the species wouldn't die out even with an asteroid impact, though that would send them back to the stone age.
So I am confident that it wouldn't wipe out Africa or South America. Famines, cancer, sure. Full wipe? No.
For comparison, the total nuclear arsenal is estimated at 5 gigatons, 80 at the height of the nuclear arms race. The Chixulub impactor was estimated to be 72 to 100 teratons and caused a 15 year winter as well as a planet wide firestorm, and it would stand a good chance of wiping humans out just like the dinosaurs.
Don't mistake this as me saying Fallout larp yolo, the northern hemisphere would be written off.
Now things would get spicy if the Yellowstone and Campi Flegrei were nuked, as those could put even Africa and SA in a big pickle. Not total extinction but stone age pickle.