On one hand many countries will invest in heavy industry, manufacturing, and fall back to fossil fuels.
On the other hand, economy overall should slow down and consumption of non-essential goods and services will drop.
I’m not expecting any definite answers or numbers, of course, just some food for thought.
In war, the economy does not slow down. It is turbocharged.
A nuclear war could counter global warming by triggering a nuclear winter but the actual effects are very uncertain. Basically, for a nuclear winter, a lot of “dust” needs to be lifted into the stratosphere. Those huge, multi-megaton bombs that they had back in the day caused a mushroom cloud that rose all the way to the stratosphere. Today, smaller, more precisely targeted bombs are preferred. It also depends on how combustible the targets are. No one is really quite sure what the climate effect of nuking a city is.
ETA: That was how climatologists saw nuclear winter ~15-20 years ago. No idea if anything has changed, but there probably wasn’t a lot of new data.
A substantial reduction in the human population would largely end the burning of fossil fuels and trigger reforestation; removing CO2 from the atmosphere. Ultimately, I would expect WW3 to greatly mitigate global warming.