IMHO, a protracted three-way cold war in the nuclear age is impossible. IOTL, MAD prevented the USA and USSR from nuking each other. They wouldn't nuke each other out of a fear of being nuked back. In a three-way cold war TL, the strategy would be different. Each power would be trying to play the other two against each other. The goal would not be to prevent nuclear exchanges, but rather to cause nuclear exchanges that are beneficial to your own power. While a cold war could develop temporarily, the constant political games would eventually result in the entire world being annihlated. If you set it before nuclear weapons, then it becomes possible but unlikely because two sides will eventually gang up on the third.
Seems like rather foolish games: once it's clear Doomsday is here and the nukes are flying, the two which are fighting are quite likely to use 10% or so of their arsenals to make sure the third won't survive to "dance on their graves."
Bruce