Has this baby seen a Certified Avian Vet since she was injured? A lot of regular and exotic's vets won't see a baby bird until they are a certain size, but any Certified Avian Vet will definitely see him for that serious injury...
****There is no doubt in the world, at all, that his "head-tilt" is a result of the head injury inflicted upon him, and it's not actually "head-tilt" at all in the sense that you're thinking of it, and this is exactly why he needs to see a Certified Avian Vet immediately! "Head-Tilt" is a sign/symptom of a neurological issue or deficit, and is usually the result of a congenital issue in baby birds. Real "Head-Tilt" is not a problem, disease, or condition at all, but simply a sign or symptom of one...
What your baby is doing is not at all "Head-Tilt" (this is exactly why it may seem to be getting a bit better, as "head-tilt" does not just start to go away on it's own)..What this baby is doing is a physical reaction to the severe head-injury he has had now for some time, AND IS VERY-LIKELY A REACTION TO SEVERE PAIN!!! Has he been to a CAV and at the very least given some pain meds, like Tramadol or Metacam? He's likely been in extreme pain since it happened, think about it, anyone would be...
****The reason that you need to get him to ONLY a Certified Avian Vet is because he needs to be seen by an Avian Specialist who knows how to administer a proper Avian Neurological exam. There are several mechanical and nerve tests that an Avian Specialist can quickly perform to identify the extent of his head trauma and brain damage, which he certainly does have, judging not only from the extent of his injury but also as obviously exhibited by his physical reaction to the trauma and pain. THIS BABY BUDGIE MAY HAVE A VERY SEVERE BRAIN INJURY THAT COULD EFFECT THE REST OF HIS LIFE, AND ALSO EFFECT HIS GROWTH AND DEVELOPMENT!
I don't know your bird breeding or hand-raising experience and education level, but a serious head/brain trauma such as this at the age of this chick is very-likely going to effect whether or not he is going to be able to learn simple tasks like shelling-seeds and eating solid foods, perching, climbing, flying, etc. Without a proper neurological exam to diagnose the extent of his injury and brain trauma, YOU'RE NOT GOING TO KNOW WHAT YOU ARE GOING TO NEED TO MODIFY FOR HIM, OR HOW TO GO ABOUT DOING THIS!!!!
I've had baby Budgies and Cockatiels who have either been injured by their parents during their first 2 weeks of life (before I would pull them from the nest-box), or have injured themselves very early-on after I had pulled them and put them in the Brooder, usually due to a larger sibling...And the results of these types of injuries at this age can be catastrophic for baby birds, as they may never fully wean or EVER learn to eat solid food, they may never fledge or even attempt to try to fly, they may never learn how to perch, how to walk, how to stand, etc. Think about a human baby or toddler who suffers a brain-injury before they get their first teeth and learn to eat solid food, or before they have ever learned to crawl, or stand, or walk...It's exactly the same situation. And this makes caring for them a life-long situation for you as his breeder, I've been there, and I had to make tough decisions with the help and advice of my CAV...
***More importantly in the immediate, you have no idea how much pain this baby is in or has been in since this happened. I am not saying this to be mean, nasty, harsh, etc. with you, but to provide constructive-criticism to you that I hope you take to heart from this day forward, because you must...
***-The minute you decide to allow your birds to mate and breed, and the minute you decide to allow fertile eggs to hatch, you have also made the decision to be fully, 100% responsible for those baby chicks in every way, including their medical care, if necessary. If you do not have the money to take both your babies and your breeding-pairs to a Certified Avian Vet at a moment's notice, then you shouldn't be breeding. Period.
*-When you are a breeder of baby birds and a baby gets injured, or one of your breeder's becomes injured or sick, you have a responsibility to get them immediate medical care from an Avian specialist. It should be no different than when you make the decision to have a human child. If you had a human baby that you were responsible for and it suffered this extent of head trauma, you would have taken your baby to an emergency room immediately, and if you didn't, you'd be arrested for child-neglect and abuse very quickly, and you'd lose custody of the baby...Think about it this way.
*-You also need to keep in mind that female birds who lay a lot of eggs, either fertile or infertile, are constantly at-risk of becoming Egg-Bound at any minute. EGG-BINDING IS 100% FATAL WITHOUT IMMEDIATE MEDICAL INTERVENTION, AND MORE IMPORTANTLY IT IS EXTREMELY, HUGELY PAINFUL TO THE FEMALE BIRD, WHO WILL BE IN ABSOLUTE AGONY FOR THE MINUTES TO HOURS BEFORE SHE DIES. So you must always be ready to take your female breeders to either your Certified Avian Specialist or the nearest 24/7 Emergency Animal Hospital at any time...Same goes for the baby chicks.
You have a responsibility to not force this baby to suffer, and I'm sorry to say this, but this is a very serious head trauma that is going to likely result in permanent brain-damage, neurological deficit, physical, psychological, and behavioral developmental issues, and a permanently disabled bird...if he lives. You need to get him to an experienced Certified Avian Vet immediately, Monday at the latest, and have them examine this baby, perform neurological testing, and counsel you on not only the extent of the injuries, but also about what is going to be the best plan of action to take for the baby moving forward...It may very well be euthanizing him, I'm sorry to say this, I've been there and have had to make this decision a few times, both with chicks and with adult breeders. It's just a part of breeding birds...the heartbreaking part. But you have to put your own feelings, wants, and needs completely to the side and remind yourself that this baby has likely been in horrible pain since this happened, and he has gotten no medical care (unless I missed where you said you already have taken him to a CAV, if so then I apologize)...
You must also remember that all birds, no matter their species OR THEIR AGE, possess an innate drive to completely hide ALL OUTWARD SIGNS OF ILLNESS, INJURY, AND PAIN. It's an innate survival instinct that they are all born with, it's the only way they survive, and the only way their flocks survive. So while you think that his "head-tilt" is "getting better", and you probably haven't "seen" or been given reason to believe that he's been in horrible agony (other than the fact that he has a horrible head-trauma that is obviously painful), he has very likely been in agony this entire time, and is also suffering from serious neurological damage... (which again is not actually "head-tilt" at all, it's simply the way his body is reacting to the head trauma and pain)...
I hope that I'm wrong, I'd love to be wrong in this situation...But that's exactly the issue, without taking him to a Certified Avian Specialist for a full exam, neurological testing, and at least a plain-film x-ray to see the extent of the damage, you have absolutely no way to know what the extent of his head/brain damage is, nor do you know what deficits and permanent brain damage he may suffer from...When I saw those photos of that baby my stomach sank...And then it sank even more when I realized that he had gotten no medical attention...So please. call your Certified Avian Vet first thing on Monday morning, tell them what happened, how long ago it happened, and that it's an emergency, and that he needs to be seen TODAY. There's no CAV that would not squeeze-in a baby bird with that head trauma, no way..At the very least he'll get pain medication and you'll get an idea of what he is up against, and what you are facing, at the worst your CAV will counsel you about what is best for him going forward...If you just let him lay the way he is now and simply "hope for the best", not only will he continue to possibly suffer, he may just pass-away, but even worse than him dying, much, much worse, is the possibility that he's going to have little to no quality of life at all. And you cannot let that happen, as his breeder it's your responsibility to make sure it doesn't happen...And I hope, as an active, current bird-breeder, that you already have a relationship with an experienced, Certified Avian Vet...if not, then again, you shouldn't be breeding without having that all set-up and in place beforehand...Just think about this long and hard, again, I can't imagine that you'd allow a human baby with this same head trauma to not get immediate medical attention from a Neurologist...why should this be any different...