As of today, there are enough known nuclear weapons as to render the world inhabitable for 98% or lifeforms, the 2% being deep ocean fauna, insects that can mutate quickly enough to adapt to the new environment and rats, who can also count on an exceptional breeding factor. And extremofiles bacteria.
A full nuclear exchange will cause a multitude of factors destined to transform Earth into a desert.
- SOOT: All main cities and military nuclear silos will be targeted first. Cities will turn into blazing infernos, adding the soot of their bonfires to what the explosions launched high in the eair. This soot will include every possible chemical subtance and further radiation if the targets (like Southern Korea or Japan, etc) used to have nuclear power plants of their own. Add to all of this the radioactive dust that ground explosion (remember the military targets) will create.
And we haven't even used ALL of those monsters. in a FS nuclear war, at this point no one wants the other to leave any future advantage. This is a perfect 'lose-lose' situation, in which each side thinks he can win for the keep.
- COLD: Nuclear winter will soon set in once dust has settled in the atmosphere. The once tropical regions will be suffering arctic temperatures. Average global temperatures will drop so fast and so bad as to surpass what humans suffered during the last great ice age when at least there was food to be found and air and waters were clean.
With sunlight severely reduced for a prolonged period, radioactive black rains all over the world, and prolonged cold, the food chain, which depends on plants, will utterly collapse. everyone dies on Earth surface, except, again, for cockcroaches, flies, and scavengers that can feed on carrions and who found themselves sheltered underground when war fell upon us.
And when, finally, the atmosphere is cleared of the dust, the ozone layer by then is depleted enough to cause a continuous, intense UV bombardment on the surface. Any god-allowed human survivor would be slowly killed by skin tumors and would get blinded.
Furthermore and even worse, those few survivors would remain clustered and scattered in small tribal enclaves unable to contact each other on long distances (no technology and no animals to use for transportation or to carry messages). Withing these enclaves, many men and women would just be infertile due to radiation. Those who could give birth would generate misshapen monsters with a very short life expectancy, and those incredibly rare babies who could reach sexual maturity in relative good health should soon start inbreeding, thus giving birth to once again defective babies. Extinction of humankind
So yes, life would survive, but it will be left to evolution and some million years before you could speak, perhaps, of a sentient species again. And this time, it will be coming from rats.