Not to freak you out.............but I always heard NOT to use Bacitracin/Neosporin on birds......or maybe it was not to use the one with the pain relief.....
I wish I could help you......Is there an aviary vet near you? Luckily I found one 1.5h away that I use in emergencies. The team there is very, very, very smart & helpful.
Is the area bleeding? How bad is the bleeding? 1st of all you wanna stop the bleeding. Use a saline solution to irrigate the area if you can.....(saline is just salt water....more water than salt) How deep is the cut? Where exactly on the neck? DO you have any butterfly bandaid things so you can approximate the cut maybe? Or, even some tape?
I would suggest building up an emergency kit for your animals. Then when you have an emergency, you will have a few things you can go to if needed.
Hoping things go well for you.............very sad when we have a hurt animal...