I'm in a similar situation with adult son, 23, with diagnosed adhd, now v low, lack of any motivation or energy, and I think suspected autism/autistic burnout. He/I are not very far into process but from being alongside my son I'd say
a. fill out the gp forms for him, to his dictation, filling in the gaps;
b. be with him in the gp and assessment appointments (with permission obviously) and
c. prep the narrative with him beforehand and
d. take notes while there. My son, left to himself in this mood, would have given so little information or so many gaps, or not remembered what said afterwards so benefits little. Liaising with the NHS on his own is beyond where my son is at right now. (If it is adhd, the medicine needs titration, - lots of dosation tweaking over time - which in retrospect I now realise was beyond my son in early days when living independently. So the assessment hard got, then failed as he failed to make the follow up online appointments to get the dosage right...)
My son has recently returned in distress to live with me for a while. I needed to be very directive to get this to happen. Is this a possibility for your son? I get they are adult, but mine had regressed, and was really at risk and not looking after himself in shared house. He seems grateful to have me being much more actively involved right now.
We did get excellent short term support from mental health crisis team, - accessed where I am by phoning 111 option 2 (I'm in Wales). Or you could look on your local nhs trust website under Mental Health Crisis Team. I'd really recommend seeing if the same is available for you, as we did get an urgent next day assessment, access to antidepressants, and 2 weeks intensive support, which did make difference then. There was a virtual ward, where you had access at home to similar levels of support you would have access to if they were sectioned, - of course only if this were an active risk, and it keeps costs and bed space down if people can be kept at home. However we are now several months on, urgent referral for actual psychiatrist isn't till Feb, and I'm now wondering about going private. It might be possible for you to talk through your concerns with this team to get their advice too, - this is something ours offered.
I'm trying (at suggestion of mental health crisis team), alongside antidepressants, to focus on a. initially getting him to eat more and wider range; b. getting him to sleep c. helping on routine, d. trying to get him to take exercise, by doing it with him. Nudging on timings; sleep routine; baths etc. Small steps but he is no longer actively suicidal.
Have recently bought a grounding sheet for beds for both me and him, and while it might be a load of baloney we are both sleeping far better. Plus a weighted blanket each, seems to be helping better quality sleep.
I've bought some books for myself so I have a sense of what I can be supporting on in the meantime. Autistic Burnout Workbook by Megan Neff is very good if autism might be in the picture. The self help groups like ADHD UK have good websites too.
My understanding (may be wrong) is that you'd need a psychiatrist for assessment (eg is it adhd/autism, prescription of medication). Is this your/his main focus? Or support now? Would he be up for / have energy for psychotherapy or counselling? If so I would choose someone who is registered with professional body (eg BACP or UKCP or others - these sites have list of people on their sites), and say they have experience with eg adult adhd/autism. If you're wondering about the difference between the different roles, and training and professional bodies Explore roles | Health Careers is an NHS careers site, which you can search for brief explanations. My understanding is psychiatrist is also medically trained (and can prescribe); psychotherapist more on talking therapy side and deals with more complex issues potentially than counsellor.
There is a local wellbeing centre in our city, which has groups like Mens Sheds and Neurodiversity peer groups. There are also mental health walking groups etc. All seem excellent ideas (for someone else) but my son not yet in place where these seem viable for him, but I mention them in case there might be something similar local to you.
I think reassuring that there is hope, that you love him and care, that he isn't alone, do all make a big difference. Plus a good meal. But I am finding it very stressful and lonely, and uncertain. And I am finding it hard to be consistent, in part because his condition is inconsistent. Good luck, and do pm me, - or continue the conversation here, if it would help. What you're doing matters.