Can you possibly take a photo of his injury/wing and post it here so we can see it? Is it still bleeding? How much total blood did he lose? Were you able to get it completely washed-out?
You have to worry about several things with a dog-bite, the first being the amount of blood your bird lost, because Cockatiels do not have much blood at all to spare, with about 2 Teaspoons or less being the maximum before death. So if the wound is still bleeding you need to put corn starch or flour if you don't have corn starch, directly on it and apply pressure. Once the bleeding has stopped, then you need to get it cleaned-out and disinfected, as dog mouths are extremely dirty and contain lethal bacteria that can kill birds. So you want to use some type of antiseptic to cleanse the wound very well...Hibiclens is best, you can buy it at any Walmart or drug store/grocery store, it kills everything and you just dilute it in a little warm water and cleanse the wound out...Hibiclens also forms a protective-barrier over the wound that lasts for about 24-hours so no other bacteria can get in and infect it...So after you cleanse the wound with Hibiclens, you don't want to rinse it out again, you want to leave the Hibiclens on/in the wound and let it dry.
You need to get your bird to a Certified Avian Vet or Avian Specialist Vet ASAP...If you can only get him to a 24/7 Animal Hospital because it's the weekend (I also don't know where in the world you live, so I don't know what you have access to) then that's fine, take him, because they can do an X-Ray to see if any bones in his wing are broken, and then you can have them send the X-Rays to an Avian Vet, and then you need to see the Avian Vet first chance you can. He will probably put him on a broad-spectrum Antibiotic due to it being a dog bite, to protect against a lethal infection setting-in.
Is your bird holding his wing in any unusual way, or is his wing at all "drooping"? If so, then that's a sign that there is an injury to either a bone, muscle, or nerve. And because it's his wing, that's why it's so important to get an X-Ray done ASAP, because the sooner it's taken care of and set/splinted if it's fractured, the better chance he'll have of being able to fly again. But infection is your main concern after a dog bite.
