Not really...the number of nukes detonated before and during the Cold War are nothing in comparison to the nukes you can drop in a game of Civ, and there was no change in the climate due to them, as far as I am aware.
I know that at the first test there was the idea it might...cause some...drastic global warming (by way of setting the atmosphere on fire), but as far as I know, no global warming OR cooling are attributed to nuclear bombs (and fewer and fewer scientists are accepting any human cause of climate change...and you might be surprised how few ever believed in it).
On the nuclear thing. Ever heard of the tsar bomb? The thing had it's power cut in half and, if buried before detonation, would have been one of the most significant seismic events of recorded history. It carried (in it's weakened form) more explosive power than EVERY bomb used in all of World War II, and the shockwave of the thing wouldn't let the fireball reach the ground and could be measured three times as it went around the world over and over.
http://www.damninteresting.com/the-most-powerful-bomb-ever-constructed/
No influence on the climate, but when you're talking about something a quarter as strong as Krakatoa's eruption, you're getting close. However, that bomb would have absolutely obliterated any city. The nukes in Civ are much much weaker....MUCH weaker, and generally more in line with the bombs dropped on Japan to the vast majority tested though the Cold War, which are totally insignificant to the global climate. No global warming or nuclear winter.
