Unless the area of bleeding is very small, an affected nail will usually fall off on its own after several weeks because the pooled blood has separated it from its bed.
A new fingernail can regrow in as little as 8 weeks, but a new toenail may not fully regrow for about 6 months. If there has been injury to the nail bed and/or surrounding tissues, the new nail may take longer to grow.
Even with the best repair, there is still a possibility that the new nail may grow back and not look normal. See your doctor if you notice any problems with the nail as it heals and regrows.