It's worth noting that only USA-Russia full "mutually assured destruction" nuclear exchange has the potential to come anywhere close to an existential threat. A limited nuclear exchange using up the arsenals of all other states does not; a nuclear escalation that results in USA or Russia nuking a non-nuclear state does not, etc. As long as USA and Russia (who between them hold almost all global nuclear weapons) keep their MAD reserves not triggered, it would simply be localized mass destruction on a scale comparable to the city/country destroying mass bombings of WW2.

It would be horrible, it would be comparable to the largest mass killings of 20th century, but it wouldn't be anything that different from what we have survived in the past and it wouldn't be a "near extinction event". "Just" a horrible catastrophe.