There is a distinct difference. Afghanistan/Vietnam are across the world this is right next door. Plus a nuclear armed threat. This time they won't care and do everything to secure the nukes and do a regime change.
Doesn't matter. MAD means the nukes hit every one involved. Pretty sure the Soviets would tell them to knock it off and the US would happily have parked nukes in Turkey like in OTL.
I have been watching south park for years. The main villain is human stupidity. Cartman is at most a vehicle to show off or instigate the plot for it but he is not the main villain. Maybe you should do a rewatch.
I will admit I'm curious in the long term ramifications of all this. I forget if you said the confederacy would win the war. Also I doubt the US would be favorable to the British if WW1 ever happens.