--unless its cell body, the soma, is completely compromised.

So axons and myelin sheathing--if you are talking about a nerve cell that actually has myelin, as most of the small fibers that subsume the sensations of pain and temperature do not--can regenerate if the offending harmful agents are removed, whatever those may be.

Myelin, though, can be replaced much faster than axonal material. There are likely no tissues in the body that regenerate more slowly than axons; an oft-cited figure is perhaps a millimeter per day of axonal growth, even under optimal conditions.