The information on WWIII in the Star Trek Universe is a little sketch and the dates don't always seem to match up. The reason is probably just due to the constant change in actual dates that come to pass. I'm not sure one can really consider the date 2026 to be the actual start of the official war. It could simply be the time at which tensions began to escalate towards war. The best information on dates is that ended around 2053.
It can also be confusing, that Spock mentions the last of our so called World Wars when discussing the Eugenics Wars. Spock later mentions WWIII taking place around 2026.
Anyway... I think that's why they chose Montana as the setting for First Contact. There's been a great deal of nuclear proliferation in the last 30 years and this may continue for another 30 years in some form or another. Montana has a lot of decommissioned missile sites, and wouldn't be a large target like Washington or New York (assuming the factions do their homework and aren't relying on Cold War targets).
For years, countries like the US and USSR conducted above ground nuclear testing at various sites. This didn't result in the entire world looking like a nuclear wasteland. It did however result in nuclear fallout being detected even decades later across those countries. They've even gone as far as to detonate a hydrogen bomb off of Johnston Island in the '60s. And I can assure you, Johnston Island was no wasteland.
Nuclear winters are still largely theoretical and no one knows what the scope of one would be. While one group of scientists say that they may be short lived if at all, the others say they could last decades. Similar theories had been made when detonating the first H-bomb... that it could ignite the entire atmosphere- worldwide, in a chain reaction.
Based on what we see in the court room in Encounter at Farpoint, other parts of the world did not fare as well as Montana... even as far ahead as 2079.
Besides... Chernobyl never stopped looking lush and green either.
One thing is for certain- at the end of WWIII, humanity was in a new dark age.