Can you have chickens and budgies in the same cage?

No, it is not safe. Cockatoos can hurt smaller birds very easily. Budgies often have "big bird" attitudes and will bother other species. The cockatoo can hurt or kill the smaller bird when it reacts to the smaller bird annoying it. The cockatoo isn't being mean or bad, it just doesn't understand why it is wrong to react to the smaller bird in an aggressive way. Even if they are bred in captivity and hand-fed/hand-reared, cockatoos are not genetically distinct from their wild ancestors. They will still act like wild animals. You can obedience-train a dog not to bother a cat, but it's not worth the risk to try to train a cockatoo to leave a budgie alone. Keep them in separate cages and never let them play together.