It is the smoke, after all (not the fallout, which would remain mostly limited to the northern hemisphere), that would do it worldwide: smoke and soot lofted by fierce firestorms in hundreds of burning cities into the stratosphere, where it would not rain out and would remain for a decade or more, enveloping the globe and blocking most sunlight, lowering annual global temperatures to the level of the last Ice Age, and killing all harvests worldwide, causing near-universal starvation within a year or two.
