I don’t agree that working on climate change is plausibly a better way to reduce the risk of nuclear war than working directly on nuclear war. Firstly, climate change is a very intractable problem in the first place for philanthropists and for national governments, given that action is opposed by entrenched interests across all society and requires cooperation pretty much of all nations. Nuclear peace is opposed by some entrenched interested in the military industrial complex, but these do not reach anywhere near as far into society as a whole. Major results could be achieved just by getting cooperation between Russia and US, which is not true of climate change.
Secondly, climate change is much less neglected than reducing the risk of nuclear war. Thirdly, there have been lots of apparently successful treaties that have e.g. limited the size of US and Russian arsenals. It just seems much easier to make progress on things that foster peace and reduce arsenals than on nuclear war caused by climate change. The path for the latter is extremely indirect.