Neurons during Numb feeling
A lower supply of blood does not mean no supply of blood, so my guess is there's generally enough to keep stuff alive for some time. But if the supply is restricted for long periods (think frostbite), then stuff starts to die.

Also, the sensory neuron endings in the skin are normally just one long projection from a cell body, the major components of the neuron are protected by being located in e.g. The Dorsal Root Ganglion.