At the risk of simplifying the situation, I think the Cold War doesn’t happen without nuclear weapons, or, if nuclear weapons had been discovered early in human history.
If nuclear weapons didn’t exist then everyone is sort of able to ignore each other following the end of the war. The United States is content to simply benefit economically from being virtually the only remaining industrial power to not have its infrastructure demolished during the war. On the other hand, Russia is content to lick its wounds and attempt to rebuild.
Both sides continue to act against each other through proxies, but realistically, it is not beneficial for either side to engage in conventional warfare, and without the threat of total nuclear annihilation, the conflicts that did happen probably don’t escalate as quickly or potentially catastrophically as they did.
As far as experience dealing with a nuclear world goes, just look at the way nuclear weapons work in the modern world. Once people deal with the looming threat of potential global annihilation for an extended period of time, everyone just sort of relaxes. No one was interested in using nuclear weapons during the cold war as an offensive weapon. However, both sides absolutely did not want to be caught by surprise in the event of an attack.
This overwhelming anxiety about acting to late or being caught unaware creates a feedback loop where the fear of being caught by surprise enhances the fear of being caught by surprise.
Once both sides realize that the threat of nuclear destruction is unlikely and that a conventional war is beneficial to no one, it greatly increases the chance that they either actively ignore each other or seek a diplomatic solution to their issues.