An increase to heart rate is usually a sign of adrenaline, along with a slightly bitter taste in the mouth, and in most people, some fine muscle tremors. I think we’ve talked about that before.
As with a lot of things, how someone handles an immediate combat threat is going to be very specific to that individual. A lot of people freeze up, and some can switch over smoothly and rapidly. Specialized training can help with this. But, it’s important to understand; this isn’t covered in most martial arts classes.
Usually training comes in two parts: First is an awareness of dangerous situations, so the combatant will be harder to take by surprise. The second part is rote responses to specific threats. This can vary pretty massively depending on who the person being trained is. It can include drawing a weapon, getting to cover, tensing muscles (which you mentioned), or going into a stance. It won’t always be completely appropriate, but it doesn’t really need to be, either. The entire point is just to get the combatant ready to fight faster. It’s worth pointing out, with military drills; those rote responses can include lethal takedowns.
How well someone handles an adrenaline rush is another matter. As far as I know, this is something that people either learn to deal with through experience or conditioning, rather than traditional training. The more adrenaline rushes someone’s experienced, the less they’ll be impaired by it, relatively speaking. In general, adrenaline rushes work towards your advantage in hand to hand or melee, but work against you when operating firearms.