OP didn't say extinction event.

That aside, if all nukes were launched, it's not about the directly irradiated areas. Imagine how much fallout will be launched into the upper atmosphere when 10,000+ nukes hit, with many in the multi-megaton range.

The sheer amount of material thrown into the air could virtually blot out the sun for potentially years. Given that a large portion will be radioactive, it could be raining death for years too.