Hair washing has been and still is a challenge. BUT we found that N is much more comfortable in the shower. He has more control over getting his hair wet (when, how, how long, etc.) and, now that he's 12, it's more appropriate for us to not be seeing him when he's doing this part of his self-care routine! He often is willing to get his hair wet (again, this was a discovery at age 11).
Not so sure how often he uses the shampoo, or what exactly he does with it (does it touch his head? Is it lathered and scrubbed around?).
I've sent my husband in a few times to observe/enforce; not as effective when Daddy is the good cop and Mommy is the bad cop.
He discovered the hand-held shower can be removed from the holder and the settings can be changed, so that was fun for him.
Good luck, and hang in there! Sometimes these kids just move inch by inch at a turtle's pace with self-care activities.

My son also strongly resists all hand-washing and baths. He loves water parks and pools and will happily jump in and swim underwater but he cannot tolerate a shower at all or any water splashing into/near his face in the bathtub; go figure! Anyway, when he was younger I let him wear his swimming googles in the bathtub because it was the only way I could wash his hair without having to hold him down at the same time. Now I put just enough water in the tub that it wets the back of his head when he lays down but doesn't come too close to his ears. As long as I am very, very careful not to splash any water in his face or ears, he will lie still and let me gently scoop water to the top and sides of his hair to wet it and then to wash the shampoo out. He seems to feel more secure/in control with that scenario instead of water coming at him from above. Not sure if any of that will help if your case but good luck!

Sorry about late reply. My son is 7 yrs and same way. What helped for hair cutting at home is that I let him sit in front of a long mirror (dollar store) and draw on it with dry erase markers or we both make silly faces at each other. This lets my guy see what I'm doing. I also have small hand mirror in case he wants. For bath time I have a shower head with different colors that light up when the waters on. All this time. I remind him that this is how Thomas the Train cleans. "Shiny and Clean" song.

Try using the swimming googles that just covers the eyes when washing his hair. Once he trusts that no soap is getting into his eyes he may enjoy the hairwashing process.

We’ve had a similar issue - 15yo/girl...She takes baths, showers are not an option. We give her bath toys (occasionally, when an extra “tool” is needed to get through her bath) & use a Munchkin bath cup to rinse her head, that flex fits to her forehead to help limit water in her eyes, also she uses a wash cloth to cover her eyes. The most useful - A MIRROR! We have one of the command strip ones, made for the shower, at her visual height while sitting in the bath, so she can see what’s she’s doing & what’s happening to her. I also encourage her get her hair wet, wash her own hair & have her rinse...all to the best of her ability, following behind her for each task to make sure she actually gets clean, but giving her a sense of ownership...mind you it’s taken a lot of patience and practice to for her to get to being so cooperative & we do still have full blown meltdown & anxiety on occasion, but bath time isn’t the dreadful experience it used to be...I’ll take 1 success out of10 tries!