This is obviously a really tricky situation and I don't think I have much meaningful advice to give. I can, however, try to give some of my perspective, as someone who has in the past been reluctant to engage with people who were trying to help me. One of the big problems for me was that anxiety/depression makes it hard to engage with anything. In situations of getting help, most of my mind was consumed by the effort of just existing with depression and anxiety and I didn't have much left over to devote to introspection or formulating answers to questions. This phenomenon is further exacerbated by the problem of having to think about my problems in order to address them. To this day, I have an instinctive physiological fear response when I see therapists or scenes of therapy (e.g. in movies) because the more I talk about anxiety the more anxious I get.
Another block I faced with this was that I didn't always feel as if I was being listened to or that my autonomy was being completely respected. I have a very hard time telling what I am feeling, but a lot of the mental health system is built on the patient being able to communicate their feelings instantly to a doctor who sometimes acts as if these few pieces of information are enough to truly understand the patient.
It sounds like your son really needs help. It also sounds like you've worked hard to get him what he needs and that you're generally aware, conscientious, and compassionate about this whole thing, so I don't think I need to tell you anything obvious or general. The best advice I can offer, based on what I've described in the previous paragraphs, is to try to engineer ways to let him take the smallest steps possible. For instance, since he came to the GP appointment, he might be willing to go to see another professional. Don't put any pressure on him at all to talk or do anything other than be there; that's a big step on its own. If that's too much, there are plenty of smaller things that would still help him and perhaps put him on the path to getting help in bigger ways; for instance, just trying to have a conversation with him about whatever he wants could be useful for him.
Here are some questions for you: Has he given any reason for refusing medication? Does he think he has a problem/want to get better?
This last thing might be near impossible, but if you can find a way, try to communicate to him that happiness is real and possible. Depression blots out the the vividness of happy memories and dulls the faculties that know how to perceive joy, so happiness begins to feel like a complete fabrication. Forcing myself to believe on an intellectual level that happiness exists (and that some will even come effortlessly as a product of regression to the mean) has got me through some difficult times.
