In Star Trek doesn't WW3 also include the Eugenics Wars where Khan Noonian Singh came to power?
The memory alpha wiki page for the Eugenics War lists the result of the war as: "Earth is devastated" and mentions humanity almost being thrown back to the Dark Ages.
There's also this quote:
This is Earth in our 21st century. Before everything went wrong. Our conflict... started with a fight for freedoms. We called it the Second Civil War, then the Eugenics War, and finally just World War III. This was our last day. The day the world we knew ceased to exist. What began as an eruption in one nation, ended in eradication of 600,000 species of animals and plants, and 30% of Earth's population. Global suicide.... - Christopher Pike
The memory alpha page for 21st century also says this in the very first paragraph:
After World War III devastated large parts of Earth placing Humanity in near-extinction
And this in the timeline:
2053 - World War III ends with a nuclear holocaust resulting in the deaths of some six hundred million Humans. Earth begins its long journey towards recovery.
There's also a whole page about the "Post-Atomic Horror", which was the era that Q recreated for his court room when he was judging Picard as a representative of humanity in the first episode of TNG.
Sounds pretty post-apocalyptic to me.