These guys are ridiculous. There are a lot of places that would likely be entirely untouched by the next world war. People who don't think too deeply are going to key in on where nukes would be falling and wind patterns to spread (low doses of) radiation and (theoretical) nuclear ashes around. While not worth ignoring altogether, you should also be worried about the collapse of consular services and economic pressures from rich countries to cause otherwise safe states to turn hostile to any obviously foreign persons. Especially if that country has a history or a longstanding fear of being colonized by people from your country or by people who look like you.

If I had to recommend a place to duck out to last minute I'd say Ecuador. Easy immigration policies, low population, moderate climate, insignificant strategically and in terms of resource wealth. They're politically uninteresting as well. They're multicultural enough that if you're white, black, east or west Indian or latino, you shouldn't have any issues blending in. Backup ideas include low population territories of greater powers. Anywhere in northern Canada, Greenland, the Faroe Islands, Iceland should give you a remote enough and sufficiently under-populated setting that it wouldn't be worth anybody's time to either attack those places, or to come there to conscript you.