Nuclear war is not an extinction event, though. Most people would survive, and while they would face hardships, the "radioactive wasteland" thing is a fairly sloppy sci-fi trope.
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.