Alas it's not . The nuclear explosion itself doesn't produce the ash and soot , the soot is produced by the material being burnt by the explosion's thermal radiation and subsequent fires . A volcano however does release ash and soot on its own .So you can't directly compare energy released between a volcano and nuclear explosions , for example the blast from a nuclear bomb often puts out fires that the radiation ignited . If you were to detonate all those bombs in the middle of the ocean , it wouldn't produce any soot . Initial theories in 1983 on the magnitude of nuclear winters relied on estimates of how much burnable material exists in urban environments (city buildings but also oil wells etc.) and made wrong assumptions on how quickly and how much it could turn into soot (firestorm vs conflagration) .

The gigantic Kuwaiti oil fires of 1991 were a good opportunity to test out various hypotheses surrounding nuclear winters , and a very important one was a "self lofting" effect that was theorized to happen once the soot was high enough to absorb sublight and lift itself up to the stratosphere where it would remain for years and cause a catastrophic cooling that could affect the entire Northern Hemisphere . This self lofting effect didn't manifest . As a result the cooling was limited to the gulf of persia directly underneath the smoke which Sagan measured at a 4-6C drop . When the fires stopped , temperatures went back to normal quickly .

Since then models have been revised and there is no clear consensus on the effect a full nuclear exchange would have on the climate (the effects on the ozone layer seem to be more dangerous and long lasting than the cooling for instance) but in general they are a lot less dramatic than initial estimates .

That doesn't mean a full blown nuclear war can't be devastating for the climate and our environment , because it would be . What it means is that nukes can't annihilate the human race .

At least not the nukes we currently have and not through nuclear winter . Several thermonuclear bombs wrapped in cobalt 59 and detonated at the right places around the globe would 100% do the trick though .