The short answer is: neurons do NOT always die quickly when deprived of nutrients. The longer answer is the following:
I am a professional researcher in brain ischemia (ischemia = no blood flow), which is the topic of the question. Armatus' answer was correct insofar as neurons have very limited energy reserves and therefore loose ATP very quickly. Gianpaolo R gave the correct time course for ATP loss. However, Gianpaolo R's explanation of cell death applies only if the ischemia lasts for about 20-30 minutes. In this case, the cells swell and lyse by a process called necrosis.
However, if the duration without blood flow (which is the duration with no oxygen and nutrients) is less than 20 - 30 minutes, then the cells do not die by necrosis. It is less well-known to lay people, but well-known to neurologists that the neurons will die days or weeks after the ischemia in a process called "delayed neuronal death".
What is also less known to lay people is that if the duration of ischemia is less than about 7 minutes (this depends on the species, e.g. human, rat mouse, etc), no neurons die, and in fact, they are protected for a short while from a similar insult. This is a phenomenon known as "ischemic preconditioning".
The explanation of necrosis is uncontroversial. However, no one knows what causes delayed neuronal death or preconditioning responses. The excitotoxicity mentioned by Armatus is a popular theory in the field, that has been largely discredited for a variety of reasons I will not go into here. It is also popular to believe the delayed neuronal death is due to a process called apoptosis, but this also has fallen out of favor in recent years.
Hence, right now, there are many competing ideas as to why the neurons die in a delayed fashion, and the simple fact is, no one knows for sure why they die in this way.
As you may imagine, being a worker in this field, I have a theory that I think is the most feasible of all. But it is a mathematical theory that applies to any injured cell, not just neurons deprived of blood flow. The theory is very simple conceptually. It says that when a cell is injured it gets damaged (call it D), but also activates genetic responses to protect itself (call this S). If the genetic protective responses are greater than the damage, that is, if S > D, then the cell recovers and lives. If the damage is greater than the stress responses (D > S), the cell dies. The rate at which the cell recovers or dies is inversely proportional to |D - S| after D and S have run their course interacting with each other.
This last statement precisely answers your questions. Neurons will die quickly when |D-S| is a large number. They will die slowly when |D-S| is a small number.
Now, this is just a qualitative description. The theory actually consists of mathematical equations that predict many things. You will note it is a true theory and does not rely on the biological specifics of the system under study. I am currently measuring the theory in my lab. I am confident it will be a very useful theory not only for explaining brain ischemia, but many other forms of cell injury.
The original article detailing this theory can be obtained at this link.