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I can’t immediately find a reference for this, but I would assume that all US forces deployed to support NATO in Europe can be commanded by the Supreme Allied Commander (SACEUR) who is always an American general. If the Russkies decide to invade, I’m sure that SACEUR has the authority to dictate response if he can’t immediately reach the President.
The only reason that NATO would need to urgently and immediately speak to POTUS is for nuclear weapons release, and fortunately that scenario has been wargamed to hell and back.
Part of the impact of that scene is that up to that point, the Death Star was not under Vader’s command, but he kind of comes in there and starts bossing everyone around anyway.