It might raise the chance of some conflicts (e.g. over Antarctica or Russia's border) and some land grabs, but I cannot see it leading to a world war.

India and Pakistan are both reliant on the Indus River for their agriculture. The Indus has its source in glaciers in the Himalayas. As those glaciers disappear, the Indus will deliver less water, and deliver it much less reliably (melting ice and snow provides a nice steady flow, runoff less so).

So: you have two nuclear powers who are both relying on the same diminishing resource to feed their people. Do you not see how that could cause....tension?

That largely depends on to what degree food production collapses in newly-tropical regions. A whole bunch of staple agricultural production isn't going to survive widespread drought and heatwaves, and everyone dependent on those food sources is going to end up very hungry, and taking a hard look at cooler neighbouring regions...