So our timeline is literally one in a million? I don't believe that. The USSR or USA wouldn't just push the big red button unless:
a. They made sure it wasn't faulty equipment.
b. A nuke actually hit their land.
The problem is that with land based missiles, the operative fact was always use them or lose them, which is why the "launch on warning" concept arose. It is really easy to construct a faulty equipment scenario that takes longer than 20 minutes to sort out. By then, the birds would be in the air, as the time from launch to impact was about 30 minutes and you needed about 5 for a Presidential order to make its way through the chain of command...
The decision making time was shockingly narrow. Assuming a massive ICBM launch detection, here's a little timeline
T=0 -- NORAD picks up faulty signs of a massive ICBM launch from within the USSR
T+5 -- Notice of said faulty launch reaches POTUS, who we are presuming is already awake
T+7 -- Bagman is summoned, opens the football and awaits a POTUS decision. Note: we are now T-23 from presumed impact and probably T-16-17 from the time a launch order could work its way through the chain of command, including a signoff from the SecDef under the two-man rule
T+10 -- Emergency plan to relocate POTUS to either an E-4 at Andrews or Mount Weather needs to be implemented, with a similar protocol for SecDef. We are now 14 to 15 minutes from the time when a launch order can be effectively implemented.
T+16/17 POTUS arrives Andrews (I've gone with the airborne NEACP/E-4 approach). POTUS now has about 13 minutes to get airborne (less, really, because Andrews and DC are presumably targets) and around 8 or 9 minutes to give the launch order.
This assumes everything goes well and that the false warning has yet to be detected. Upon arriving at Andrews, the time for a decision is essentially at hand. I find it very easy to see this going bad, with the faulty warning not being picked up before a decision is made in an atmosphere of shock, haste and panic. Note that a false detected sub launch off the US coast would shorten this timeframe considerably...