That depends on when the attack came. General Issa Pliyev, the Soviet commander in Cuba, was very assertive, and throughout the crisis he interpreted his orders as aggressively as possible. He had launch authority for his tactical battlefield nukes and cruise missiles (which could easily destroy Guantanamo and the Marine flotilla off Cuba's northern shores) but not his ballistic missiles. On October 22, he got new orders not to use any nukes. A war started at that point might have stayed conventional, although that seems supremely unlikely.

(Castro, by the way, was convinced the Americans would launch a pre-emptive strike and demanded several times that the Soviets launch their nukes first. In any shooting war where the Soviets didn't use their nukes, it's likely Castro would have sent Cuban forces to try and seize them.)

On October 27, a Soviet sub captain almost fired a nuclear torpedo at a US destroyer enforcing the embargo. That definitely would have led to nuclear war. The same day, Pliyev's air defense teams shot down an unarmed American U2, killing the pilot.

At this point, Khrushchev issued new orders to Pliyev. He would avoid any further provocations but he was given ambiguous orders about launching his nukes.

Khrushchev was a hardened veteran of bureaucratic war. Since his orders on nukes had been crystal-clear to this point, and since he knew Pliyev was champing at the bit, the message is obvious. Khrushchev did not want to be the one giving the final launch order, but he wanted to ensure that if the Americans launched a nuclear strike they would pay a terrible price for it.

The Americans intended to stage a conventional invasion of Cuba, but they planned massive retaliation against any use of nukes - and they didn't know Pliyev had tactical nuclear weapons. Given Pliyev's orders and temperament, and Castro's eagerness to hit back with nukes even if the Soviets refused, it's likely any limited war would have escalated very quickly into an unimaginable catastrophe.