Climate cause and effect is difficult to nail down, but it seems with that much firepower back then a "nuclear winter" scenario was much more plausible. We have dismantled a lot of the nuclear arsenal since the 1980s these days -- estimates I see of current firepower vary by quite a bit, but I'm not even sure there's 6,500 megatons of nuclear weapon firepower in service globally anymore.
In one article in the New York Times from the 1980s (http://www.nytimes.com/1984/12/14/opinion/in-the-nation-tamb...) the estimate used for exploding half of the world's nuclear arsenals is equivalent to 6,500 megatons of TNT.
Mount Tambora, in contrast, a much more powerful explosion than any 20th century eruption, is widely quoted from various sources as being equivalent to 800 megatons of TNT (example: https://books.google.com/books?id=NuFfCgAAQBAJ&pg=PA15&lpg=P...). And Mount Tambora definitely had climate effects (https://en.wikipedia.org/wiki/Year_Without_a_Summer).
Climate cause and effect is difficult to nail down, but it seems with that much firepower back then a "nuclear winter" scenario was much more plausible. We have dismantled a lot of the nuclear arsenal since the 1980s these days -- estimates I see of current firepower vary by quite a bit, but I'm not even sure there's 6,500 megatons of nuclear weapon firepower in service globally anymore.