WW3 almost certainly also means no more communications satellites and no more weather satellites, in addition to no more humans in space.

It would likely also mean nuclear winter and many other bad things, but permanently denying use of satellites is likely the first global cost of WW3.

Nuclear Winter has been generally discredited by modern modelling. The whole premise was likely politically motivated by scientists who wanted to make nuclear war less likely to occur, by dissuading politicians with this yarn. (A true noble lie.) Criticism of the hypothesis was muted by few scientists wanting to risk the perception of supporting nuclear war.

As for "permanently denying use of satellites", this is only partially true. High orbits would be fucked for millennia, but LEO would mostly clear up before the century is out. I know that seems 'forever', but it isn't. Also, depending on the severity of the debris, interplanetary exploration would not necessarily be out of the picture; if the debris is severe enough that a GSO satellite has a 99% chance of being destroyed within a year, then that orbit is as good as ruined. But if you only plan to pass through that region for a few hours, not spend a whole year there, you might be able to do that if you can tolerate some risk. It'd be like running across a highway, except the cars move faster than bullets and won't brake for you. If you hang out in the middle of the road you are certain to get hit. If you time it right and run across, you have a chance of getting through.

If you go back a bit farther, I think you'll find the work on nuclear winter actually came out of work done at Cornell to try to model the result of a large object impacting the earth 65 million years ago to kill off the dinosaurs. Amazingly enough, one of the hardest parts of the dinosaur extinction work was figuring out how a localized impact like that could kill enough things to cause an extinction, and that part came from the atmospheric modelers. After doing that modeling work, they realized the parallels with a nuclear conflict and produced the nuclear winter projections.

I think there's little doubt that tons of soot and ash in the upper atmosphere can cause periods of global cooling. The problem with the nuclear winter hypothesis is in the link between nuclear war and tons of soot in the stratosphere. These nuclear winter models make a lot of assumptions about the likelihood of a modern cities turning into firestorms and the amount of soot they would generate, the amount of soot lifted into the stratosphere by the fireball, and the amount of soot that stays there instead of precipitating out.

Then it turned out everything was cooked, not frozen.