Considering an accident that results in heavy bleeding (say legs are crushed), but no internal damage to the brain or torso, what would stop first: the brain or the heart? Is there an order to organ failure from loss of blood?
It's actually not terribly complicated. The heart will stop first.
In the scenario you describe, the heart will continue pumping until it runs out of blood to pump, and it may even continue pumping activity after that even though there's no blood to move. This is called pulseless electrical activity (PEA), previously known as electromechanical dissociation (EMD).
After blood flow to the brain drops below below levels sufficient to maintain consciousness, the person will fall unconscious. The brain will not be dead but injury may begin occurring. Once blood flow to the heart drops below levels sufficient to keep it functioning, it will stop completely (asystole) or go into ventricular fibrillation, both of which mean no blood is being pumped and it will be quickly lethal if not treated immediately. At this point, the brain will begin to die.
The time difference between when the heart stops and brain death occurs is measured in minutes, commonly estimated at about 8 minutes. But the answer to your question is that the heart will stop pumping before brain death occurs.
According to internet searches there seems to be no known order. There are no empirical studies to be found using animals, and the issue thus appears a theoretical one.
In casual understanding the question might refer to the point in time when the heart and brain shut down forever, i.e. when there is no more possibility of resuscitation, which roughly translates to times of cell death in brain (brain death) or heart respectively (no defibrillation possible any more). The second part of this answer deals with this situation, because:
In much better understanding the questions appears to target the times of the stop of functioning („shut down“) of both organs. The terms "excessive bleeding" in a less evident sense seem to imply that the bleeding must not be thought of as being stopped by help in order to maintain some specific volume of blood for longer times, nor the effects of blood loss healed by resuscitation. These circumstances must be excluded to answer this question.
The latter situation does not translate to cell death but to
Cardiac arrest of heart opposed to entry into coma of brain
To paraphrase the question: All other organs excluded except brain and heart, what is first with extreme blood loss: cardiac arrest or coma?
About cardiac arrest see answer to related question on StE: "By definition, cardiac arrest means the heart is not pumping blood."
About entry into coma to compare (in time) to cardiac arrest see e.g. related question on StE: "... loss of consciousness from a lack of blood supply to the brain and the treatment required after..." Cardiac arrest and coma are both reversible by treatment, however, the question refers to untreated continuous blood loss.
Sheer logic might answer the question whether coma or cardiac arrest comes first, as by loss of blood the heart, different from the brain, "loses its job": there is no blood to pump, no substrate of function. However, there are no empirical findings to this simple logic to be found by search.
According to basic knowledge, massive loss of blood leads to
Hemorrhagic or hypovolaemic shock
See e.g. Medlineplus:
"Hypovolemic shock is an emergency condition in which severe blood or other fluid loss makes the heart unable to pump enough blood to the body. This type of shock can cause many organs to stop working."
"Hypovolemic shock is a dangerous condition in which your heart can’t get your body the blood (and oxygen) it needs to function. This happens because you’ve lost a large amount ― more than 20% ― of your blood volume. (…) While your body fights to meet its demand for oxygen, it makes your brain and heart the top priority for receiving oxygen. (…) Stage 4: You’ve lost more than 40% of your body’s blood (more than 2,000 mL or 68 ounces). Your blood pressure is low and your heart rate is high.", Cleveland Clinic
From what is known about hemorrhagic shock, cardiac arrest is the first effect of blood loss, and cardiac arrest may lead to coma. It is not the other way round:
"The loss of coronary perfusion pressure adversely affects myocardial contractility; cerebral blood flow decreases, resulting in the loss of consciousness, coma, and eventually death." Clinical review: Hemorrhagic shock
The mechanism of hemorrhagic shock implies that it is not coma that comes first, inducing cardiac arrest. It is cardiac arrest that induces coma. This causality may be hidden or not explicitly stated as in that situation coma is equivalent to "instant" death. Conversely, again, coma is not the cause of cardiac arrest in hemorrhagic shock.
With blood volume vanishing and decompensation impossible, the heart enters into the status of fibrillation that precedes cardiac arrest. Fibrillation can occur in hemorrhagic shock, but not always does. This may be a matter of "excessive" speed of blood loss, the wording of the question may hint at. There may be no time to switch into fibrillation mode. Considering, on the other hand, slow speed, trickling out while internally bleeding, the question does not appear to inquire about fibrillation not being able to prevent the brain from coma. Fibrillation may not prevent coma for a long time; however, that's not what the question's wording "excessive bleeding" seems to address.
There may be different grades of oxygene saturation for the heart muscle to avoid arrest, and for the brain nerves to avoid coma. However, my internet searches brought up no data that would specify any saturation rates according to different amounts of blood volume, differentiating brain from heart. There seem to be no empirical studies on excessive blood loss, different oxygene saturation rates corresponding to different volumes of blood.
Hence, the answer is: heart first.
Second part of the answer:
What about cell death as irreversible cardiac arrest/irreversible coma?
To tell from what is known about survival of heart muscle and brain nerve cells in anoxic condition (lack of oxygene), the answer should go:
See, e.g. Medline: "Brain cells are very sensitive to a lack of oxygen. Some brain cells start dying less than 5 minutes after their oxygen supply disappears.(...)"
Conversely, heart cells seem to be able to survive much longer, and reference on what may be profane among professionals cannot easily be retrieved from the internet; from what I found it can be inferred that muscle cells are much less prone to damage in anoxic situations i.e. lack of blood supply (muscle cells are known for anaerobic muscle contraction). I found the following scientific paper from which it may inversely be inferred that heart muscle cells can survive for hours, Ganote/Van der Heide, "(...) Control and 45-, 75-, or 90-minute anoxic hearts were exposed to hypotonic (200 mOsm/l) perfusion media to induce osmotic swelling. (...). See also, e.g., Goodyer Left ventricular function and tissue hypoxia in irreversible hemorrhagic and endotoxin shock, 1967, and Kara Marker, Labroots: "Even when oxygen levels reached just ten percent, heart cells continued to produce ATP. They do so by applying a process that doesn’t need oxygen to produce ATP. In this way, heart cells can maintain normal function without adequate supplies of oxygen, but they can only do so for a limited amount of time." Nerve cells seem to more sensitive to a drop in blood supply, with no possibility of resuscitation.
Interestingly, the question refers to the fact that the heart, by pumping blood, supplies oxygene to the brain and, at the same time, to itself. Thus, again, sheer logic might answer the question.
If the heart died second it would be the brain's function to take care of the heart supplying oxygene (by blood, according to premise of reasoning). However, it's just the opposite.
Funny as this argument may appear, it does contradict and does not confirm the answer given. From sheer logic the answer were: Heart first.
This paradox cannot be logically solved by referring to the brain supplying vital nervous input to the heart. The phenomenon of brain death shows: the heart may still beat when brain has "shut down".
The mechanism of shock, as the question interestingly brings to the point, in my opinion might have evolved to safeguard the brain's function because the heart in a developmental and evolutionary sense did depend on specific brain cells ("selfish heart), and only in pre-coma situation shuts down first.
StE Question related question: What happens to the heart during cardiac arrest
Bonanno, Hemorrhagic shock: The “physiology approach”