Once all the sites are operational (which would be sometime before the turn of the year), Khruschev is going to announce it to the world. The Soviets intended the missiles on Cuba to be part of their deterrent, and there isn't much point in a deterrence that your opposition does not know about as Doctor Strangelove helpfully reminds us.
So the crisis literally can not begin any time later then 1962. Of course, if Khruschev is able to get all the missiles he wants operational, then there probably isn't any crisis: the missiles are already there and operational, everyone knows this*, and the US has no way to get rid of them without an unacceptably high risk of getting nuked. It's a fait accompli at that point.
*IOTL roughly 6-8 missiles, along with ~100 additional tactical nuclear weapons, were already operational when the crisis began. The US didn't find out about this until well after the crisis (and by "well after" I mean 1992) and the Soviets never told them for some reason. It is interesting to speculate on how the crisis might have gone if Khrushchev had publicly said early on that there were operational missiles: the whole point of the blockade was to prevent the Soviets from getting in enough stuff for that to happen and such an announcement would make the US realize the blockade had technically failed even before it began.