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 where 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?
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 captain, the political officer, and Arkhipov. The B-59 was being depth charged by US ships in an attempt to get it to the 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 flotilla 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.
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.
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.
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.
Had Khrushchev rather than Lieutenant General Stepan Grechko ordered Anderson shot down...
Had Kennedy failed to correctly guess that Khrushchev hadn't made the decision to shoot Anderson down...
The war would have/could have started.
To back AlexP up, it may in fact be very difficult to start an all out nuclear war of the type often envisioned.
Take a look at this list of nuclear close calls. At least two of them were prevented by only one person, and others were prevented by only a handful. So – what, are we just getting lucky or something?
The truth is, it's really hard to start a nuclear war.
The reasons for this is many-fold. Nuclear weapons are greatly feared, especially by those involved in actually using them – and there are a lot of people involved (from the president, to generals, to pilots, to missile crews, to submarine captains, to the comms officers relaying orders, to the individuals actually pressing the button, and many more) – and any one of those people could prevent at least one launch. In a sense, firing a nuke must be agreed upon unanimously by everyone in the chain of command from the president to launch technician – any single 'no' will result in the delay or prevention of a launch.
There's a famous study called "Men Against Fire" written by a well respected military historian that indicates that only about 15% of soldiers actually:
when ordered to. Now, there is some debate about the study's methodology, but when you think about all the people involved in using nuclear weapons, even if that study is overly conservative, there's a high likelihood that many perfectly functional nukes would not be fired.
The study implies there's an 85% chance that any one person in that chain of command will say 'no' to a launch order. If there are just 3 people between the launch order and the nuke itself, you're looking at only a 0.3% chance of a launch. And hey – maybe the situation is different for nukes, maybe American military discipline is enough to overcome that in this case, or maybe individuals won't say no as long as they're not directly responsible for the launch. All the same, the odds of a full nuclear launch being successfully executed by all involved are pretty slim.
Moreover, with all the false alarms and the 70 some odd years without a nuclear holocaust, some generals complain that many missle crews wouldn't be prepared to retaliate in the case of an actual nuclear attack.
All of this is to say that if the US or Russia ordered a full nuclear attack of their entire arsenal, only a minority of their weapons would actually be fired. So the world ending in 3 to 7 minutes seems unlikely
That said, and to answer the question, the key would be to have the crisis escalate gradually. It's much easier to order a launch when there's actual fighting happening. It's much easier to launch a nuke when your home town has already been hit.
American intelligence indicated there were at most five nuclear missiles in Cuba. In reality, there were scores. I think the simplest change would be to have had the Americans escalate slowly with either conventional airstrikes or an invasion force (as many generals were recommending). We know Soviet commanders on the ground were authorized to use tactical nukes, so once they felt they would be overrun, it becomes much more likely that they would go nuclear.
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 İzmir. 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.
You might want to look into a movie titled The Bedford Incident, made in 1963. In this movie, a US sub chaser and a soviet sub mutually destroy each other, but cooler heads prevent an escalation to full scale war. And of course there's Doctor Strangelove, where a full scale war does break out.
In the actual crisis, the soviet sub did not fire a tactical nuclear weapon, as @Andon pointed out. The firing of that nuclear torpedo might well have triggered a series of escalations for two important reasons. First, the "nuclear threshold" would have been crossed and the US would certainly have considered a retaliation using tactical nuclear weapons. Second, an attack on a warship or merchant ships has signaled US entry into war in the War of 1812, Spanish American War, and World War II. Those two psychological factors could easily have signaled the beginning of a war.
Also, you have to consider President Kennedy's public announcement that any missile launched from Cuba would be treated as a direct attack by the Soviet Union on the United States, requiring a full retaliatory response.
And finally, you might want to look into the post mortem analysis conducted jointly by US and Soviet officials as to what might have happened differently. I don't know when this was held, but I know Robert McNamara was part of the US team. Both teams concluded that we came very, very close to all out war.
There are a total of four known incidents that occured during the Cuban Missile Crisis that would have sparked a Nuclear War. The most likely was the sub incident detailed by @Andon discussed.
Also during the Cuban Missile Crisis, a series of Bombers, armed with Nuclear Bombs, was staged at a small runway at a U.S. Air Base when an emergency alert started the scramble of the planes. The Alert was soon identified as an intruder alert, not an incoming attack alert as the planes were almost ready to take off... the crisis was averted by an air-force personel driving his jeep onto the run way in the path of a plane while the situation was stood down. The intruder was later identified to be a bear playing with the AFB's fence, setting off the alarm.
Later that same night, a routine polar recon flight flew off course after the pilot was disoriented. He had become turned around and flew into Russian Airspace while trying to get correct co-ordinates from his base. His radio alerted him to a Russian Music station broadcasting as his first sign of trouble. Russian fighters were scrambled for intercept but the lost pilot was able to make it back home. None other than President Kennedy was briefed on the situation the next morning, and he was not at all happy with the pilot.
Finally, the U.S. early warning systems were tipped off to an attack inbound from Cuba during some point early during the crisis. The problem was found to be with the calibration of the EWS's new directioning. Prior to the knowledge of missiles in Cuba, the EWS was pointed towards to north pole, as it was the shortest flight path between central Russia's missile fields and the States. These systems were shifted to look southward when Cuba became a problem in great haste and calibration issues occurred. The New Jersey station detecting the inbound launch were scrambling out notices to important people when the missile started drifting over the Atlantic and away from the U.S. Turns out, they did not detect a Cuba originating launch, but a NASA launch from Cape Canaveral. In all the confusion, no one in the government thought it was a good time to tell NASA to delay their rocket launches for the time being.
One of the most serious but less-known trigger-points during the Cuban Missile Crisis was the fact the Soviets had approx 100 tactical nuclear weapons already deployed on Cuba.
Soviet commanders on the island did not need permission or launch codes from Moscow to fire their weapons, and were already positioned near Guantanamo with instructions to defend in this way.
American military leaders were unaware of these. During the crisis, military leadership argued repeatedly for a full-scale marine invasion of Cuba -- for which they had already drawn up plans, and were eager to execute.
An invasion by the Americans would have been met with a nuclear strikes on Guantanamo Naval Station and American fleets. With invasion fleets destroyed, the only action remaining to the Americans would be an immediate nuclear counter-response.
As mentioned in other answers there were also several other potential trigger-points: a nuclear-armed submarine under attack, a spy plane shot down, Castro's request to the USSR to launch a pre-emptive strike, as well as the naval blockade.