Can nuclear missile subs set their own targets arbitrarily?

If thats the information you get, then say your prayers, you are already dead. Its like getting information that a guy is pressing the trigger on a handgun, whose muzzle is contated with your temple; 0.025s before the action cycles.

Mobile launchers can fire within a minute from pre-surveyed sites.

For the U.S. at least we still have the concept of the “nuclear triad”, including manned bombers which can launch not only nuclear-armed cruise missiles but also drop nuclear bombs. Those would presumably be the weapons that would be used in any kind of “Godzilla is attacking Los Angeles”/“Martians have landed in Iowa” sort of scenario, rather than trying to re-target our own SLBMs or ICBMs to hit American soil.

Of course, even supersonic airplanes are vastly slower than long-range ballistic missiles; so as AK84 points out, trying to preemptively take out a mobile missile launcher that way (or any other way, most likely) is fundamentally unworkable. Bottom-line, both superpowers spent a lot of time and money and effort making damn sure that the Other Side could not somehow “shoot the gun out of their hand” and then simply dictate terms, thus “winning” a nuclear war (and maybe “only” killing a few tens of millions of people and conceivably still leaving an enemy country worth conquering).

Already the return of great power competition is making people wonder if the reduced Arsenals are an invitation to strike first.

Part of the problem in retargeting missiles in the subs is that they form part of the strategic weapons component of the nuclear defence. Their job is not to strike enemy forces engaged in belligerent or threatening action. Their job is to retaliate and render an attacking enemy’s country incapable of defending itself against even a band of kids armed with pointed sticks. The targets for this are determined a long time ahead, and don’t change. Even if you got into a conflict that was starting to escalate badly, the nuclear subs would not be called into play. The subs are there for ultimate retaliatory action, not first strike, or tactical strikes. As such there isn’t a requirement that the missiles be retargetable. Prudence would suggest that targeting would be managed in much the same manner as for land based ICBMs. A launch order code that selects from a set of targets, none of which are known, or knowable, to the sub’s crew.

An interesting (to me) aspect of nuclear command and control is the E-6B Mercury aircraft which relays the commands to sea and land assets

In some ways, the interior of the aircraft is informative. People who believe the government has secret technology years or decades ahead of the civilian world - and sure, in limited domains it does - would wonder why they are holding back the good stuff for the aircraft to be used in Armageddon. Where’s the antigravity or holographic displays? Or even capacitive touch screens or VR?

Francis Vaughn, that’s exactly why they must be retargetable. You don’t know what the remaining important targets are going to be, after all else is said and done. Maybe a few of the initial ICMB targets survived. Maybe the most effective strike to reduce the enemy’s capability will be against mobile assets. Heck, maybe World War III was against a completely different country than you anticipated.