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My story takes place centuries in the future, in the year 2567. Over 600 years ago, during 1962, an all out nuclear war broke out. Most modern day historians believe it to have occurred in autumn, but whatever happened, it caused the United States and Soviet Union to go into nuclear war, destroying everyone, and everything, in a RAGING INFERNO OF ATOMIC FLAME

You and I would refer to this event as the Cuban missile crisis

My story takes place in an alt-universe were the CMC went hot, and all out nuclear war commenced. I want to make my story as grounded as possible, so my question is, what is the smallest change needed to turn the CMC into a nuclear crisis?

Not Much

Or, more specifically, one person making one decision could easily have done it.

That person is Vasili Arkhipov. During the crisis, he was the second-in-command on a Russian submarine, the B-59. On board this ship, there had to be a unanimous vote to use nuclear weapons - The captian, the political officer, and Arkhipov. The B-59 was being depth charged by US ships in an attempt to get it to surface. With no communication, and thinking they were being attacked, the captain wanted to fire a nuclear torpedo. The political officer agreed. Arkhipov, who was also commander of the submarine floatilla and who had gained stature in the K-19 incident earlier that year, voted no. He was well-respected and also of equal rank to the B-59's captain, which allowed him to convince them to do otherwise.

If he voted to agree, they would have launched a nuclear torpedo. It would be extremely easy for the US to retaliate and for things to escalate.

Simply because one man made one decision.

Reflecting on @Andon's answer: One man's life could have done it.

Major Rudolf Anderson, Jr.

During the Cuban Missle Crisis Major Anderson was flying U2 recon missions over cuba. Despite being famously told in the movie 13 Days to not be shot down by special assistant to President Kennedy Kenneth O'Donnell, he was shot down.

And died....

From the above History channel link...

For Kennedy and Khrushchev, Anderson’s death crystallized their realization that the crisis was rapidly spiraling out of their control. “It was at that very moment—not before or after—that father felt the situation was slipping out of his control,” Khrushchev’s son Sergei would later write. Kennedy worried that retaliatory airstrikes would inevitably result in all-out war. “It isn’t the first step that concerns me, but both sides escalating to the fourth or fifth step and we don’t go to the sixth because there is no one around to do so,” he told his advisers.
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That night, the president dispatched his brother to meet with Soviet Ambassador Anatoly Dobrynin and offer a top-secret deal to peacefully end the standoff. The Soviets agreed to remove their nuclear missiles from Cuba, while the Americans pledged to withdraw intermediate nuclear missiles from Turkey and not invade Cuba. The tensest moments of the Cuban Missile Crisis had ended, with Major Anderson the only combat casualty in a standoff that had the real possibility of killing millions. ...

“Anderson’s death escalated the crisis significantly,” said Upcountry History Museum historian Courtney Tollison. “It could have provoked a cascading series of events that if you follow to their logical conclusions lead to a nuclear World War III. Instead, his death was a jolt to Kennedy and Khrushchev and pushed the crisis to a point where they had to take one of two paths, both of which had clear consequences.”

Had Anderson not been shot down, or had he been taken alive as a prisoner, the crisis could have easily turned to all-out war.

Alternatively...

1. Had Khrushchev rather than Lieutenant General Stepan Grechko ordered Anderson shot down...




2. Had Kennedy failed to correctly guess that Khrushchev hadn't made the decision to shoot Anderson down...

The war would have/could have started.

In 1961 the American Empire deployed one squadron of 15 PGM-19 Jupiter nuclear-tipped medium-range ballistic missiles in Turkey, at bases around the ancient city of Smyrna, which the Turks call Ä°zmir. This was perceived as an aggressive maneuver by the Russian Empire, which at that time did not have any effective means of defending itself against an attack with medium-range ballistic missiles.

In reponse to the American deployment of MRBMs in Turkey (and Italy), Russia started a very visibile program of deploying a number of R-12 Dvina (which the Americans called SS-4 Sandal) in Cuba, with the capability of hitting most American territory; they also embarked on a program of preparing launch sites for their intermediate-range R-14 Chusovaya (aka "SS-5 Skean" in American materials), which, if deployed, could hit targets in all the contiguous territory of the U.S.A.

The American leadership realised that the time window when they could have started a nuclear war with a reasonable hope of success had closed. After a few days of tense negotiations, which, among others, resulted in the establishment of the famous Moscow–Washington hotline, the two superpowers agreed to dismantle their advanced missile bases -- Russia took back its missiles from Cuba, in exchange for America taking back its missiles from Turkey.

The truth is that by 1962 in was already too late for the U.S.A. to launch a successful nuclear war against the U.S.S.R. The Americans had had their opportunity in the mid-1950s, when they really had overwhelming nuclear superiority; but it so happened during those few years when America could have indeed won a nuclear war with Russia, they had Dwight D. Eisenhower as president; Eisenhower did not want war, so there was no war. And then the window closed, and neither superpower could hope to start a nuclear war and win in any meaningful sense.

But what about 1962? What could have happened in 1962 to precipitate a nuclear confrontation? In real life, nothing. The Americans never even considered a nuclear response, and the Russians knew very well that they could not hope to launch a nuclear attack against the U.S.A. and survive. But in fantasy?

In fantasy one could imagine the members of Kennedy's EXCOMM falling prey to the panic which permeated American media, or being co-opted by those congressmen who agitated for a muscular reponse. But I don't think that even a nuclear attack on Cuba would have made the Soviets retaliate with nuclear strikes against the U.S.A. An attack on Russian targets would have been necessary; how to make the American national command authority lose its head and order missile launches against targets on Russian territory is left for the storyteller to imagine.

The good news is that at that point in time the two superpowers did not yet have the capability of killing everybody on Earth; while most interesting parts of the U.S.A., western Europe, and European Russia would have been destroyed, most of Asia, all of Africa, Central and South America, and Australia would have escaped with minimal ill effects. Great point of departure for an alternate history novel sequence.