How would it feel to bleed to death and how long would it take?
It depends on whether the bleeding is arterial or venous. If an artery is severed, the rate should be increased (although occasionally an artery will spasm and close itself down quite a bit). In the case of an injured artery, the blood would squirt out with each pump of the heart. After you lose about one liter of blood (an adult has about 5 liters) you experience signs of shock.
Venous bleeding is usually slower, described as "oozing."
Remember that bleeding can be external or internal, but the same things apply. For example, internal bleeding can be caused by bleeding in the gastrointestinal tract (esophagus, stomach and gut). External bleeding can be due to a traumatic injury.
If the big blood vessel coming off the heart (aorta) is punctured by a knife or bullet, the bleeding would be fast, because it is like a high pressure hose and the muscle making up the wall of that blood vessel cannot "clamp down" very much. If the same injury occurred in the heart, the muscle making up the heart can help partially close up the wound, so that it takes longer for the injured person to "bleed out." Some people have been known to run several city blocks with such an injury to the heart after being attacked.
If an injury occurred to the external surface of the body, there are pain receptors in the skin that would certainly register severe pain. After some time, maybe the body's own pain substances (endorphins) might kick in. Also, if one was involved in doing certain things (for example, fighting in a battle), all of the adrenalin pumping might cause the person to overlook their own injury.
After the initial injury, the person would get dizzy and "woozy." This would be worse when standing then when sitting or lying down. The pulse starts to get weak and "thready", because the heart tries to keep the blood pressure up. The person starts to feel anxious; something is very wrong. Eventually when the blood pressure drops below some critical level (which is going to be a little different for each person), the heart starts to go into an electrical disturbace called a cardiac arrhythmia.
So, if a simple small blood vessel was cut (especially a vein) and pressure or a tournaquet was not applied (for example the person was unconscious and nobody was around to help), it could take many hours to bleed to death. If you had a severe injury to a critical artery it could take just a few minutes.