So sorry op
I'm not far from Medway myself so will share your Facebook appeal to local groups that haven't had it posted already. I hope now Kent live have picked it up that Kent police will feel they need to be seen to be doing more.
You say on Fb that there was a possible hospital sighting of your son, but need more Police support to investigate? Do you mean like accessing hospital records and cctv footage? I would definitely get the police to class your sun as vulnerable (his past mental health as well as ADHD surely put him in this category?)
Have you tried bbc south east today news and Meridian? I've seen many local missing persons reports featured before and they have quite a far reach in the south east counties. I don't know what the local tv news service for Luton is but it would be worth contacting them as well. Do you have friends as well as family members who could help with this kind of thing? As I know it's going to be very time consuming and only so much you can do yourself in any given hours, especially when you're out on the ground searching in Luton.
I would get as many young people you know involved with things like sharing tiktok posts, his friends perhaps? It has a huge reach with young people, so much more so than other social media platforms like Instagram, Facebook, Twitter etc. Things can go viral and become a frenzy though, with people turning amateur sleuths, that could have it's negative sides if it hampers any formal investigations. On the other hand, just having the appeal info and photo of your son shared around by people in Luton on tiktok could help with any possible sightings locally. I know from my daughter when a local teen went missing recently they were sharing the information around and it spread like wild fire on tiktok, Snapchat as well actually. There were concerns about the young girl being recruited by county lines, it's becoming more and more prevalent unfortunately. I don't know the full details but thankfully she was found, and there were mental health issues involved in why she disappeared.
You say your son's always on his phone at home and likes tiktok, can you access his account? Same for things like Discord or other apps he may have used to communicate with people. His search histories, emails etc. as others have said could all help with information, I don't know if your internet provider or the app / web companies themselves could give you any information or access if the police aren't looking at them? Someone up thread mentioned the websleuths website, I agree there will be people including professionals on there who could help with knowledge about tech companies and accessing your son's online history, you can also post an appeal on there too xx