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:
- fire their weapons,
- at the enemy,
- with the intent to kill
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.