I don't know much about clicker training with birds, but I'm a very experienced dog trainer, and use clicker training there. The time between the "click" and the food reward depends on the scenario. When you are first building the association between the clicker and the food reward, you want to simply click and toss/give food immediately. This is so that the animal understands "click = food is coming". Once the association has been made, the time between the click and the reward can be expanded. Done properly, I have dogs that can wait 60secs for a reward and still understand what behavior was marked. The clicker means food is coming, though it may not be immediately. Eventually you can have the animal perform more behaviors before actually rewarding with food. So, the "3-second" rule may apply when you are first teaching the animal the association between the clicker and the food reward, but doesn't necessarily apply once the association has been made and the animal understands the clicker is marking good behavior, and that a reward will come later.