We won't see a true world war until the effects of climate change really start to cripple the west.
Then it will be a scrap to horde as much of the worlds resources as possible.
Nations like the United States, with large, mostly uninhabited, northern territories will survive global warming with no real problem.
Lab grown meat, and the coming green revolution will make up for food demand. We will also turn to eating a lot of bugs.
The south will be a desolate place in 150 to 200 years but before then it is going to be pretty beautiful as the increased amounts of rainfall and flooding have actually ended long standing droughts.
Case in point, Central Texas. Were I live the summer is usually pretty brown, but due to the volatile weather this year is the most beautiful year that I can remember (and I have been here for 31 years). The massive amount of rainfall has saved Lake Travis (was down 50+ feet and in danger levels), replenished ground water and kept the countryside green.
I am not sure how long this bad weather will remain but most climate change models seem to show an uptick in volatile conditions so hopefully for a few more years (get flood insurance Texans!).
The great migration north is coming though. I already have been eyeing Montana for retirement. I think 30 years from now, my retirement age, will be the right time to abandon Texas.
The interesting thing will be how the US deals with people trying to migrate from countries who don't have multiple climates (Central America). I predict a giant wall be built in about 50-100 years manned by drone troops and mounted machine gun turrets to keep people from trying to to cross the Great American Desert.
Global warming will kill billions, but the US and Canada should be ok. I have no idea how smaller countries with one climate will survive though.
I am more worried about automation, class based civil war, a rogue asteroid, and/or the singularity killing us off.