There doesn't exist anywhere near enough nuclear weapons to kill all humans. Even in the 1980s, people were predicting between 20 and 100 million immediate American deaths from a full-scale Soviet attack, and since then there are much fewer nuclear weapons. See e.g. the discussion in this comment thread: https://slatestarcodex.com/2015/10/31/ot32-when-hell-is-full...

It has been suggested that humans recovered from a population bottleneck of less than 30,000 individuals (the Toba catastrophy). A nuclear war would leave billions alive.