Not really, a vast majority of the nukes are pretty small, delivering about 10-50 kt or so. Tsar Bomba is a compete outlier. 16,000 nukes would cover an blast area of only 150,000 km2 or so, which is only 0.003% of the Earth's surface area.
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Well, it's the fall out I worry about not the actual blast.
Again, it'll only cover a very small area. Many cities will be destroyed, for sure.