The fight or flight response is instantaneous- it is a reflex that is triggered at the moment you sense danger and are fearful. So as soon as you sense danger or you are fearful, then your amygdala sets into motion your body's survival response, and this occurs almost immediately, where your muscles tense up, your heart beats faster and stronger, your awareness of your surroundings sharpens, your vision becomes hyper-focused on the threat, your breathing increases, your liver produces more glucose for fuel, and your sweat glands excrete more sweat to cool you off. All of this body response serves to increase your survival rate, and it occurs instantaneously, when it is needed. There is no advantage in having a survival response if it does not occur immediately and instantaneously. It is the lightning-fast speed of this survival response which increases your chances of survival.
Please see the following video which explains the fight or flight response, that occurs almost instantaneously: