Signals are transmitted from one neuron to another by using special chemicals, called "neurotransmitters". The speed of these signals depends on how fast the exchange of charged ions is inside and outside of the cell membrane. The main ions involved are sodium, potassium, chloride, and calcium.

Without going into details, I can say that messages in the brain can travel at speeds up to 268 miles/hour.

In order to learn more about this process, please go to the next site, which I found not only very didactic but also very interesting.

The speed with which nerves transport impulses is being studied by many people. Communication with the brain and how the brain processes the received information is still very mysterious. But scientists have an idea how fast nerves send signals. It varies among different animals and humans, but in general one can say it is very fast, on the order of 115197 ft/sec (3560 m/sec). A larger time component is the delay between an impulse and the actual transmission of that response by your nerves. And that is what nerve damage is about. If a nerve is damaged it takes longer before the nerve transmits a signal.

This is an area that will be under investigation for years to come. You are asking a very important question.