The speed of light is pretty darn fast
So when the Soviet premier tells the president that he hears explosions in the northeast, and THEN sees bright lights, it is completely backwards. The light and incredible heat from the blast (the real killer) travels almost instantaneously and this would have killed him before any sounds of the explosions got to him.
You can see this effect in old films of nuclear tests from the 50's. I remember one in particular where the camera is focused on a house. A bright flash comes from the left of the screen and the paint on the house vaporizes into smoke, a moment before the blast wave blows it to smithereens; not even enough time to exclaim wtf was that!