I cannot speak for russian military doctrine. As for the Americans, trident subs (the ones with nuclear missiles), exist primarily for strategic deterrence. A trident is parked of the coast of a hostile country and waits for further orders, nothing more. Because they run off of nuclear power, they are only restricted by how much food they can carry, which is a 3 month supply. As far as actually using their missiles, most scenarios put them at the forefront of a first strike attack or a retaliatory attack immediately afterwards. Nearly all scenarios from the US side focus on preventing nuclear war, and if that's not possible, ending it as quickly as possible. These subs would only hide underwater at the start of a nuclear war if told to, and there are no plans that I know of that specify this action.

Furthermore, it is now impossible for American subs to fire nuclear missiles without specific orders. At one point captains technically had the authority to do this, though this was taken away over the possibility of a rogue captain launching missiles.

In summary, it may have been technically possible for Americans in the first part of the cold war to do this, but it was not a part of military doctrine. It is now impossible for American subs to do this now.