Marginal to significant decrease (in "probability of a positive singularity"). The recovery would monopolize global attention and effort.
ETA: The probability of "all-out nuclear war" between India and Pakistan is very low (and predictions of a nuclear winter will make a negligible difference to the odds of it occurring). Do you know the difference between strategic and tactical nuclear weapons? A strategic nuclear arsenal is there to deter other nuclear powers from destroying you, by giving you the means to destroy them in return, even after your own nation is in ruins. A tactical nuclear weapon is one which gets used on the battlefield. For example, Pakistan would potentially use nuclear weapons against the Indian army, in the event of a massive invasion. There are very strong incentives against escalation from tactical nuclear exchange to strategic nuclear exchange.
A strategic nuclear arsenal is there to deter other nuclear powers from destroying you...There are very strong incentives against escalation from tactical nuclear exchange to strategic nuclear exchange.
You seem to assume a very rational evaluation of the situation on both sides, even given a highly escalated situation like the destruction of the opposite army by means of nuclear strikes. If the people who run Pakistan and India are that rational, doesn't that mean that people who are able to design AGI's, capable of undergoing recursive self-improvement...
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.