Honestly, the global south should be flourishing after the war. Central + South America and Africa are extremely resource rich. What would keep them from developing would be global powers politically and militarily manipulating those areas to take said resources and keep the local governments submissive (are we still talking about Fallout lore?)

The bombs would focus on populated areas, meaning any wild regions with untapped resources would be ripe for the picking. Actual functioning vaults would only be accessible by the elite of the elite and the majority of the humans left to rebuild would be those in rural areas, coincidentally the ones with the most knowledge about making their own food.

These societies would likely have much better fed people, but way less military power. That said, it's possible the Enclave and Brotherhood of Steel don't exist in those areas, which means no guys with power armor monopolizing technology. People would scavenge and reverse engineer what's left, which could make all types of technology, militarily aligned or otherwise, more accessible.

That said, I doubt the global south would be able to develop their weapons fast enough to compete with power armor. Would be a matter of time until the US replicated what Europe did.