Another nitpick: there's a difference between a nuclear-powered submarine, and a nuclear-weaponed sub.The latter are always the former, but you'd send the former to drill for oil and dig up mammoths, not the latter. They need to be in the right position to launch (so heading off to the South Pole on your own decision is going to get the captain court-martialed unless the USSR branch of UNIT speak up for him).
And the Prof must have been in the KGB to have a walkman and cheek the military the way he did: which isn't actually a problem, the KGB were in the vanguard of reform in the early 1980s, as they knew the reality because they were trusted with western info, not the propaganda (they just didn't expect reform to go as far as it did...).
Oh, and aside from the idiot who thawed out the ice block, it wasn't an ice block, more an ice box that the Ice Warrior could punch its way out of, which makes no sense at all.
And... but I don't care, I liked it!
Well, to nitpick your nitpick a little
, the USSR initially did build a diesel-powered class of submarine carriying ICBMs (the Golf-class).
Also, i'm pretty sure, this sub isn't going anywhere soon (or anytime), AFAIK sowiet and russian subs use soidium-cooled reactors which are permanently in need of heating the sodium to an liquid state, when the reactor is shut down and there is no external source to heat the sodium, it solidifies and the reactor is permanently destroyed.
Also, when surfacing through ice, the turret-mounted diving planes need to be put in a vertical position to prevent damage to them, so they are most likely nonfunctional too.