My best guess is that they were nuked out of fear to rise and take power after the other countries were weakened. It wouldn't surprise me if foreign troops stormed America in fallout 4 or later. They might even make a fallout that takes place in a countries that wasn't hit but the economy fell after the collapse of the rest of the world. This is just a guess since they don't talk much about the subject.
This is one of the reasons I always say there won't be a nuclear holcaust. I mean who's gonna nuke Nambia, Mongalia, or the thousands of tiny barely habited polynesian islands?
Greenland, Iceland and many of the 'boring' European countries were affected by fallout from the Chernobyl disaster in 1986, and they'd suffer a similar effect, amplified many times over, in the event of a global nuclear conflict. The scientists in Antarctica, without access to their usual resupply drops or personnel rotations, would die of starvation. And to the answerer above me, I assume you're referring to Mongolia? Check an atlas or map of the world and you'll see it shares a long border with China, one of the main protagonists of the Great War, who would suffer so many nuclear impacts that Mongolia would effectively be sterilised by the resulting fallout.
Countries like Greenland would have felt a drastic impact. The only real source of food for Greenland is fish and, considering the oceans are filled with radioactive substances, the food source would have dried up. Coupled with the immediate after effects of the great war could have drastically changed the eco-system (whether it be dramatic cooling or rapid heating).
During the events leading up to the Great War with the fuel crisis and so on all minority countries were being annexed by the superpowers (i.e. Canada was annexed by USA) it can be assumed that the same can be said for these countries you speak of whether it be by the Republic of China or any other super power that was around at the time of this event.
Well the nations/outposts you speak of arent self sustaining, and the global nuclear dust and other such thingeès would further inhibit their ability to survive, especially since food was running short even before the great war. Anarchy and desperation won't help either.