I'd much rather it be females but I suspect most possibly the guy with the most guns and most willing to use them will persevere.
And while women may have an advantage by being more likely to work together I suspect they will be screwed over by trusting men who them let them down, betray them or just plain use them.
Women on their own I think would do just fine. Definitely better than just men.
PS: nuclear winter won't wipe out all life, not even all human life. Even the worse predictions (which have been mostly disputed) only forecast an extended and fierce winter, not a "Snowball Earth" where the entire planet is frozen from pole to pole with glaciers in the Sahara desert. See, as much soot a nuclear war might release into the atmosphere, far more long lasting is the greenhouse gasses that are emitted, the soot will fall relatively quickly. You'll have a very long and harsh winter (for both hemispheres) then some seriously bad global warming.