If the Nukes were all on one side? No. Assuming that Germany, Italy and Japan all had large nuclear arsenals as well? Then I'd say that there's a reasonable risk that we might not be here today. Or assuming that the parties to the alliance system that lead to WWI were nuclear armed, I'd still see a good likelihood of WWI opening anyway.
Just to point out, HAVING atomics doesn't lead to MAD. Knowing what atomics can do leads to MAD. If Hiroshima and Nagasaki hadn't eaten mushroom clouds, other cities would have. We've never been able to even conceive of having a weapon capable of ending civilization completely before, so some sort of nuclear exchange was inevitable, going by the old logic of bigger explosion=better weapon. If WW2 had atomics from the get-go...Luftwaffe and RAF nuclear-armed raids, mushroom cloud over Pearl and a follow-up Doolittle on Tokyo...
On the plus side, everybody left in that world would be too terrified of nukes to ever even contemplate having them after all's said and done.