Luckily we have more than one nerve cell connecting each area of the body to the brain. However if all the nerve cells are damaged then the brain can no longer send messages to the hand. This is paralysis!
A friend of mine also cut her arm really deeply, and cut the nerve in her forearm. Her hand actually was paralysed! It took nearly a year for her to get full feeling and control of all her fingers back, I think. So try to avoid broken glasses, is the moral of the story!