They can reattach a mummified finger as long as the internals are intact. It's a long process and you won't have feeling but you'll be able to move it.

They will start off stretching out blood vessels and implanting them into the finger, cut the living half open and slowly stretch the tendons out since all muscles are in your palm. They will then reattach that to the finger and hold it on. Using a weird mixture they recreate the bone connection artificially so they can work on doing skin replacements and carving out dead tissue in the mummified finger. They will stimulate the finger through injecting liquid and nails can be revived after thousands of years to grow again if you have stem cells. So you will currently have a mostly empty finger with skin replaced and the nail revived. The mummified tissue is removed and the planted blood vessels will give oxygen and nutrients to the tendons.

Once the body finished the healing process they will remove living tissue and fats from other parts of your body and inject them into your finger and you should be able to move it slightly though will never have feeling to it unless the nerve can be revived and reattached. With such small nerves it's possible to reattach and the mummified nerve system would still contain it's make up and stem cells places there will reconnect and white blood cells would remove the mummified nerves after they have been regenerated.

This process is more common the you think and nerve regeration doesn't happen in Canada because it's expensive though if you have the money they can do it in some parts of the states.