A related medical term is neuropathy, which refers to nerve damage or disease. Dealing with a subclass, mononeuropathy stemming from, say, carpel tunnel syndrome, the page observes:
The damage to the nerve can result in numbness, tingling, unusual sensations, and pain in the first three fingers on the thumb side of the hand.
Neuropathy could be other places besides the hands and feet. Other subclasses of neuropathy from Diabetic neuropathy symptoms are:
- Peripheral neuropathy (most common, affecting feet, legs, hands, and arms)
- Autonomic neuropathy (affecting the autonomic nervous system)
- Radiculoplexus neuropathy (affecting thighs, hips, buttocks, or legs)
- Mononeuropathy (affecting a specific nerve)
If you are referring more to the sensation (as opposed to a medical condition), a different medical word referring to the tingling is paresthesia, which are described as
a sensation of tingling, tickling, pricking, or burning of a person's skin with no apparent long-term physical effect
So a patient might walk in to his doctor's office, and describe that he has tingling in his right hand. The doctor would place in the EMR that the patient reports paresthesia, and then run some tests and find out that the cause of the paresthesia is mononeuropathy caused by too much typing and mousing in EL&U.
(Even though you're looking for a medical term, English speakers will colloquially say their leg has falling asleep when sensation has left the limb in question, usually because the circulation was cut off. There is no parallel leg is waking up, as circulation returns and one senses pins and needles or tingling.
The French would say Fourmillements dans les mains ou les pieds ("ants in my hands or my feet"). (Note the fourmis (ants) that @FumbleFingers referred to.) I think that fourmillements would be a considered a colloquial term, while paresthésies would be the medical term.)