The lack of ICBM capacity for either side makes nuclear weapons in the hands of Pakistan and India effective as MAD deterrence due to the simple fact that any use of such weapons is likely to be nearly as destructive to their own side as it would to the enemy. Nations simply do not engage in suicidal behaviors. Not intentionally. Not even excessively religious groups.
That being said:
Suppose that an all-out nuclear war between India and Pakistan were to occur and were to result in climate change killing 1 billion people.
So you're postulating that there would be a global -- outside-of-India/Pakistan -- death-toll of 1 billion persons? Well, I can see that regarding China... but as to the impact of "a positive singularity"; it seems that there would be a negligible impact on this. The areas where most research and development is ongoing for these possibilities are outside of the primary impact zones for this sort of thing. We already grow far more food than is actually needed to feed the planet; reducing the demand for food by nearly 25% would only make that problem far less significant.
In fact, the long-term impact would probably be beneficial if we assume the direct correlation of CO2 to GMT modulation is as the IPCC tells us; it would prevent both China and India from outstripping the US as primary CO2 contributors.
Edited to ad: In this I was mostly just following my from the sub-post I was responding to in the general context of the OP, I sort of took the bit and ran with it, it's not a really honest response to the original question.
Your first paragraph indicates that you may not fully understand the sorts of weapons systems that are available. I don't have up to date information, and if I did I probably wouldn't be allowed to share it, but there is a class of weapons called "theater ballistic missiles" that includes a sub-category of "Short range ba...
Last month I was involved in a conversation thread about what the impact of a hypothetical nuclear war would be on existential risk.
There are many potential nuclear war scenarios which would have varying impacts on existential risk. It's difficult to know where to start to gain an understanding of the long-term of nuclear proliferation.
For concreteness, consider the case of an India-Pakistan nuclear war.
According to Local Nuclear War, Global Suffering by Robock and Toon,
[...]
Note that this would presumably cause some degree of chaos in the developed world.
I have not yet investigated the credibility of the papers' claims. However,
Suppose that an all-out nuclear war between India and Pakistan were to occur and were to result in climate change killing 1 billion people. Then would the probability of a positive singularity increase or decrease and if so why?
This question seems very difficult to answer; maybe altogether too difficult for humans to answer. I welcome responses raising relevant considerations even in absence of a good way to compare the relevant considerations. Please read the linked conversation thread before commentating.