Thank you for reaching out to Ask Lafeber.
It’s difficult to say if the nail will grow back or not. It would depend on if damage was done to the nail bed. If it was just the tip of the nail, it should not be a problem. If its’ closer to the toe it may not grow back. If the nail bed was damaged it could grow back but often times it may not grow back normal. This is a situation I would consult with an avian veterinarian about. There is one concern though. If this bird is going in and out of a nest with a mate, are they currently taking care of eggs or babies? If this is the case, it could be detrimental to remove the bird from the cage to receive medical care. A rule of thumb is if a pair of birds are set up to reproduce, they should be left completely alone other than changing their food and water and lightly cleaning their cage. I’m not sure why the owner was trying to catch the bird in this situation. However, at this point the bird is showing obvious signs of discomfort and may need to be on some anti-inflammatory meds to eliminate pain.
It would be helpful to know if this is a breeding pair of birds or if they are given a nest to sleep in. If it’s just for the birds to sleep in, I suggest removing it from the cage right away. This will only create nesting or other unwanted behavior. If I can assist you any further, please reach out.