sounds like phantom pain, but I'm not sure if it effects partial digits like it does entire limbs, but I don't see why it wouldn't. Basically, your body is constantly sending neural signals back and forth, expecting a reply--"all systems check" sort of thing, which has an expected delay when the limb in question is idle. People that have lost limbs often suffer from phantom pain where, when the signal does not return, the brain interprets this as the limb being in use for some purpose. Say you are missing a hand, the brain will adjust to the delayed signal and think "Oh, it is a fist," and you get the sensation of a clinched fist. Unfortunately, since you can not un-clinch a fist that you do not have, this sensation persists and is difficult to remedy.
Painkillers and other drugs can work, but of course don't solve the problem and are usually quite addictive.
One rather cool, and very successful treatment has been to place the remaining limb in a mirror box, have the patient stare into the box and look at the reflection of the limb for a set amount of time, over a period of several days. Whatever the treatment schedule. Actually seeing the reflection, as if telling your brain that you have two limbs instead of one, tends to trick it into releasing those signals and alleviating that pain. I think it even manages to remove the accumulation of those signals over time.
FYI: my granddad lost his leg killing Nazis in Normandy (crashed his glider on D Day), and so he had a wooden leg for as long as I knew him, and dealt with phantom pain his entire life. I was still young when he died (11, I think), so didn't really appreciate all that he went through because he kept the drinking and pain killers away from us, but this is something that he dealt with for 50 years. However, treatment for this has progressed well beyond that and has become more sophisticated. Basically: yes, it can persist and might be a permanent issue, but I think there are treatments. At the same time, severed nerves can be very tricky. It's hard to predict how or if they will fix themselves.
I seem to recall there being some effective stem cell treatments for severed pieces of digits in recent years.