Alright, so let's break this down. In our environment, we have the following elements definitely afflicting the individual:
- Exposure to desert conditions
- Blood loss (For the record, a liter of blood loss could be considered severe. It's not enough to induce hypovolemic shock, but it is enough to cause a slew of problems like weariness and dizziness and poor coordination.)
What we want is to find an appropriate mix of these elements (along with some other potential elements) that will cause this person to become unconscious for an extended period of time -- say, 12 hours. In other words, we're putting this person in a coma.
These sorts of things tend to cause bouts of brief unconsciousness, but not a state of comatose, unless anything is particularly severe. The type of extended unconsciousness that I believe you're looking for can only be achieved through some sort of injury done to the brain. Even with normal unconsciousness, what's really happening is a malfunction in the brain's neuron system.
If we break this down, there are three main ingredients that we want to look at: Dehydration, blood loss, and physical exhaustion. Desert conditions lead to dehydration which is the primary concern, and hunger in the short-term only serves to hasten fatigue and doesn't pose a major threat to an individual.
Most living beings are majority water, and need a regular intake of it in order to survive. Of the three listed conditions, this one is the most life threatening when taken in isolation. For mammals, water carries important nutrients throughout the body, and helps regulate internal temperature.
Dehydration alone doesn't cause comas, though, until it becomes severe. Before you experience extended unconsciousness, you will experience fatigue, delirium, hallucinations, nausea, muscle spasms, and organ failure. If you want to go through the dehydration route, then you're introducing a lot of risk. Once someone falls into a coma from dehydration, they are at serious risk of death. Since this person is in a desert, this is a potential route you can take, but it's also a very risky one.
The loss of a liter of blood in humans equates to a class II hemorrhage. The symptoms of blood loss at this level are surprisingly similar to the symptoms of moderate dehydration: lower blood pressure, dizziness, cold skin temperature due to vasoconstriction, accelerated heart rate. However, no unconsciousness. If you want to knock someone unconscious purely from blood loss, you have to venture into the territory of life-threatening injuries.
This is the route you probably want to take, but it's not a perfect mold of what you're after. Push someone in the desert, force them to do periods of physical exercise for several hours on end without any food or water, and injure them a bit. They may slip into unconsciousness every now and again if their body simply can't handle the exertion, and they'll be deficient on several layers of coordination, but they won't be totally unresponsive.
This sort of stress can cause memory loss and poor capacity for movement. This is probably as close to what you want as you're going to get.
Note that I'm not a medical professional, nor am I a biologist. I believe most of what I said is accurate, but if you want to be concrete with the accuracy of your worldbuilding, you may want to refer to other sources.