It will take years for the beak to grow longer and he will probably need a vet to help reshape the growth to make sure it does not overgrown on one side and undergrow on the other.
I had a cockatiel baby maimed by a lovebird while she was still in the nest. Took off Diva's upper beak almost to the cere. Her parents abandoned her, of course, and I ended up handfeeding her for 18 months till she completely weaned. Diva was 11 years old when she died in a tragic accident, but her beak never grew more than a quarter inch long. Luckily it was long enough she could pick up food and swallow it and she was able to groom her feathers. She could not hull seed or chew on wood or anything like that and lived mainly on Harrisons HiPotency Fine and finch sized Roudybush as well as soft foods, well cooked pasta and grated veggies. She behaved like a normal tiel, climbed in her cage and interacted with the other tiels. Diva had a very good life and I miss her terribly.
I also had my Senegal attack a cock cockatiel and bite of the curve of his upper beak. Benny slowly grew back the curve in his beak but I had to have my vet trim the growth every three months because it was uneven. The entire curve didn't grow back, but it was still growing when he died of kidney failure last year.
My vet said Diva's beak did not grow much because of where it was injured. Benny's growth was a normal part of the beak which is constantly growing thoughout the life of the bird and that is why his seemed to be regrowing. I don't think your bird's beak will regrow much, but he still will be able to live a full and happy life.
i am wondering if the little Black Headed Caique i have chosen will regrow it's beak and how long it could take. the Breeder has never experienced this before and does not know. s/he is about 9 weeks old. s/he kinda smiles 
I have a couple more pictures if i need to post. He said it was a recent injury. s/he must have gotten their beak stuck in cage bars and gotten scared. I also noticed that this baby perches and uses the beak just like my other caique does. I am worried about my other older 5 month old, who has a full beak but has some leg abnormalities, and whether she could or even would hurt the newer smaller BHC baby. Please let me know what you all think. Thanks! Im sure it goes without saying i fell in love which is why im asking
