If no nukes are involved, the US and its allies would have won easily.
Soviet military supremacy is a total myth. People seem to forget it was the Allies who, via lend-lease, returned Soviet Union from certain death and made it much more powerful than it would actually be without the help from Allies. For instance the US supplied the vast majority of Soviet's trucks, allowing them to have way more strategic and tactical maneuverability and better logistics than the Germans, who relied on horses throughout the war.
People tend to forget you don't need nukes to destroy cities. The Allies didn't need nukes to destroy Tokyo, Dresden, Hamburg, Berlin, they did it via carpet bombing. Keep in mind by 1960s, jet bombers have way more payload than ww2 era bombers. NATO would wipe out major Soviet cities via conventional bombing just like they did to Germany and Japan.
NATO army had better organization, command, strategy, tactics, equipment, weaponry, like they always did. Soviet army is much like Imperial Russia, gigantic and seemingly powerful, but just like WW1 and WW2 showed, these don't mean everything. Germany singlehandedly defeated and almost defeated (if not for lend-lease) Russia during TWO world wars and NATO, with the full force backed by the US, stands no chance of actually losing the conventional war.