Let's assume that 50,000 nuclear devices are launched. Note that this is itself probably an overestimate; by definition the first targets are the weapons themselves, and anything that suffers a launch delay probably never gets fired at all. Furthermore, there have been few if any actual live tests of ICBMs; this alone means that failure rates will be high. Quite a lot of warheads will be damaged by the stress of launch and reentry and fail to explode, or not explode properly, or miss targets by sufficiently large amounts that they are basically harmless. And because of this, some high-priority targets would be double or triple targeted - and two explosions in the same place don't do as much damage as two separated explosions.
And from this we see something else: much of the megatonnage - perhaps 20 Gt for the US - was concentrated in a small number of warheads. Basically, you have a few hundred 'citybuster' multi-megaton warheads in each stockpile, and thousands of smaller warheads.
Again from here there is the effect of localisation. Some areas of the northern hemisphere would be disproportionately targeted (i.e. The USA, Russia, and Europe, plus possibly China. But southern hemisphere countries like the whole of Latin America, India, Africa and to some extent Australia would be relatively unscathed.
Fallout from nuclear weapons is surprisingly light, especially at distance. 435
This is because by definition, as little Uranium and Plutonium as possible are used; these materials are expensive and heavy. Essentially, if you were in much of the southern hemisphere and not next to a military base owned by the US, you might not notice.
Even in heavily targeted areas, if we assume a 5km average radius of destruction, that's 80km2 per bomb, or 4 million km2 of complete destruction for 50,000k explosions. That's half the land area of the US.
There would be a 'nuclear winter' effect.. but we have already seen this with volcanic eruptions. We would see some global cooling for the 2-3 years it took for the particulates to settle out. For wildlife, this would be no more than the normal effects of a couple of bad seasons.
For the heavily bombarded and radioactive zones, we can compare to the Chernobyl area. After some initial destruction from acute radiation poisoning - generally localized - The removal of humans appears to benefit wildlife far more than increased background radiation harms it. Expect forests to rapidly regrow in the destroyed areas.
So, how does this answer the question? Western civilization as we know it would be devastated; many - perhaps most - people in North America, Europe, Russia and China might die either directly or from famine/disease in the aftermath. After a coupe of years of disruption, wildlife would start to thrive and reclaim the ruins of the cities and suburbs. Human civilization would probably continue in the southern hemisphere, and recolonize the North after a few decades.
It is also quite possible that between failures, counter strikes on weapons and over-concentration on military targets, the destruction would be much less than feared and most countries would still exist in some form. 435