Humanity literally physically came together because it was necessary to rebuild.

I'm pretty sure that didn't really happen until after the Vulcans showed up, TBH.

From Memory Alpha:

During the 2060s, Cochrane and his team of engineers began developing the warp drive. (Star Trek: First Contact) The challenge of inventing warp theory took Cochrane an extremely long time. (ENT: "Anomaly (ENT)") In 2061, he was responsible for Earth's first successful demonstration of light speed propulsion, though his work was far from complete. (VOY: "Friendship One"; ENT: "In a Mirror, Darkly, Part II" library computer file) His primary motivation for commencing warp technology was financial gain in the devastated, poverty-stricken America that existed in the wake of the Third World War.

He finally built Earth's first warp ship, the Phoenix, in the hope its success would prove profitable and allow him to retire to a tropical island filled with naked women. A historical irony was that, contrary to the fact he went on to use the Phoenix to inaugurate an era of peace, Cochrane incorporated a weapon of mass destruction into its design; he constructed the Phoenix in a silo on a missile complex and used a Titan II missile as his launch vehicle.

(WWIII ended in 2053; First Contact was on 5 April 2063)