How do you treat a nipped tail?

If its not an open sore or bleeding wound, just leave it heal naturally.
I had a catfish with ich. He had 2 small bloody sores, I left him alone, in a week it was healed, I almost saw it healing before my eyes, better every day. A healthy fish will heal fine. If he's being nipped or stressed by other fish he's afraid of, this could delay healing. Remove the nipper or the nipped one, I'd say. If in a week its not better you could try melafix.

I have a betta and half his tail was missing one day. I believe it was a filter accident. He was in a divided tank with another betta. I put in bettafix (like melafix for bettaS) for a couple of days. then I moved the betta to a community tank (not because of the tail), stopped beta fix, and just let it do it's thing. It is now fully regrown. this happened in sept/oct, and just recently I stepped back and realized that his tail was back. I have a molly with a small bit missing, and I am just going to leave her as long as she looks and acts healthy.

Neither fish was nipped though- both a fairly aggressive if confronted- so you should remove somefish- as stated above.

Let it be for the time being. If you start to see signs of infection (red, black, or white around the edges, or anything fuzzy or slimy starting to grow....or the fin receding) then you should treat with meds. You may want to try something mild first, such as Melafix. If that doesn't improve the wound, switch to a broad spectrum antibiotic such as Maracyn Plus. Your best bet is to keep the water very clean while it heals