There were a number of officers with launch authority.
Many Soviet subs carried nuclear torpedoes as well as SLBMs; at one point during the crisis (accounts vary) a captain ordered his sub's torpedoes armed and fired at a US Navy ship that had been harrassing his submarine. His first officer persuaded him not to launch. In the event of a full-scale war, the Soviet subs could have launched their missiles even if Moscow was incapable of giving orders.
Nuclear missiles in the Soviet Union and Europe were more tightly controlled and would have required direct orders to fire; again, in the event of a full-scale war, they would have been able to act independently.
General Issa Pliyev, the Soviet commander in Cuba, was also under this standing order; his ballistic missiles (with 1-megaton payloads) were under central control but he had authority to deploy tactical weapons (12-kiloton warheads for his cruise missiles and bombers) if he was attacked.
When the blockade began, Moscow revoked Pliyev's launch authority, terrified of escalating to a full nuclear exchange. Pliyev moved to combat readiness just in case, rushing deployment of his missiles and moving the warheads close to the launchpads.
On October 26, Pliyev told Moscow he would use "all available means of air defense" if attacked. The Kremlin approved this but again warned Pliyev not to arm his missiles with warheads unless specifically authorized.
On October 28, Pliyev received new orders, allowing him full control over his defensive actions. That meant that at the height of the crisis- a day after Pliyev's AA crews shot down an American U-2 and a Russian sub nearly nuked a U.S. Navy vessel, Khrushchev OKed the use of nuclear weapons to react to a conventional assault. Castro, although out of the nuclear command loop, was convinced a sneak attack was coming and urged Pliyev and Moscow to mount a pre-emptive nuclear war. Luckily, he was ignored.
The next day, Khrushchev ordered all nukes in Cuba stood down.
By the time the crisis broke out, Soviet units in Cuba got nuclear warheads for tactical and strategic missiles and nuclear bombs for the medium-range bombers, IL-28, a total of 162 units. General Gribkov, who participated in the work of the Soviet headquarters operations, said the commander of Soviet forces in Cuba, General Pliev had the authority to use them in the event a full-scale US invasion of Cuba.