How many atomic bombings have resulted in stratospheric soot injection? The answer, I believe, is 0, even including the two atomic bombs that were dropped on populated areas. Major conflagrations that result in stratospheric soot pumps are extremely difficult--the Kuwaiti oil fires didn't manage that, and that involved literally setting oil wells ablaze uncontrollably.
Without stratospheric soot injection, there's no global climate impact except on very short terms. Even with stratospeheric soot injection, well, look at how severe the global cooling following Pinatubo was: we'd reverse global warming since 1850... for a year or two.
(Like GP, though, I'm not keen on actually testing this hypothesis in real life.)
How do you know that soot injection and nuclear winter are the only possible side-effects from detonating all the bombs at once? We are still frequently surprised by experiments in chemistry and physics that produce weird results when scaled up. The Quantum Hall effect, for an example off the top of my head.