What time period are you considering "Cold War?" I mean, that's moving from 1945 - 1990... A huge sweep of history, and if IIRC, they adjust as the ability of nuclear weapons to be delivered swiftly and precisely changes.

Basically, the two-man rule applies to Presidential orders involving nuclear weapons all the way down to the guys in the silos turning the keys; the National Command Authority is the President together with the Secretary of Defense, who must confirm the order and issue his own code to be authenticated. If the Pres is looney, the SecDef would be within his wheelhouse to say, "No, not doing that."

There's also the provisions of Article 4 of 25th Amendment, which sets out the conditions the Cabinet and Vice President can temporarily declare the President incapacitated.

There was a case, which you can read about in this rather supercilious article from Slate, in which a Air Force major assigned to missile duty asked this very question. He was denied promotion and was severed from duty.

When you say "president" do you specifically mean JUST the United States?

What if France (first test 1960), UK (1952), Israel (1979), went "insane" and tried to nuke USSR or a country within its sphere of influence?

What kind of checks and balances were there?