Here's the short answer: We probably could not survive a nuclear winter. But the long answer, well, it depends on which countries are going to war, how many nukes are being dropped and where those bombs are being detonated.
Life Noggins explains that in a simulation with 100 nuclear bombs being dropped between Pakistan and India, black carbon would rise to the stratosphere and absorb sunlight. This would increase the temperature up there, while decreasing the temperature on the surface of the Earth and in the water of the oceans. The ozone layer would get screwed by this too, which would screw over our farms (and ourselves) and could lead to famine.
After five years, the temperature on Earth would be 1C lower and the ozone layer would be messed up. Which doesn't sound like the worst thing in the world, but remember this nuclear winter only started after a 100-bomb fake war. There are 15,000 nukes on this planet. Things would get much, much worse if all of them were used.