Hey there. I was in a situation last year when I started my first course where I didn't have any diagnosis. Before I go into everything, I will say that I am now diagnosed formally with autism, social anxiety disorder, AFRID and depression.
From experience and my friends, it's so difficult because each Uni works completely differently. My university were so amazing to me but they were also very blunt about what they could and could not offer. So they gave me deadline extensions, smaller exam venues, the option to tell tutors, and an attendance waver. However what I really needed was study support. They admitted that it was a funding issue and they could only do so much because DSA was supposed to do the rest. Some universities don't even accept self-diagnosis. Personally, I appreciated their honesty to me and they've been amazing in that regard. But everything is SO much more difficult without a diagnosis, it's horrible.
Sadly the option that helped me would be a lot harder for you because of what you described. Gps can diagnose social anxiety and write you a letter for it (though I charged me for it, it was worth it) because I was able to get DSA. That DSA gave me a councillor, assistive software. However, I had to struggle through first year without study support, which I'll now be getting next year. So that was a half-route for me.
My friend was more in your boat. She can't go to the dp because of her anxiety so she didn't have that option like I did. She ended up seeking a councillor from the University, who was able to provide her with more help than the disability adviser. They emailed her tutors with her permission to let them know the situation. Is this something your university has?
While I was waiting to be diagnosed, I was in contact with our student disability network who we all just shared our experiences. I tried some support groups to help with the anxiety, they didn't help me much but my friend still goes to them today.
Another option is if you have a tutor that you really trust to tell? I asked for a meeting (well I'm not going to lie, I asked my friend to write the email) with the head of my course. We sat down and I told him about my problems and the issues with waiting list. In my case, he was understanding and we have a support system where I can let someone know if I can't attend class and need work sent to me.
Have you asked your disability adviser if the uni offer any assessment for different disorders? My friend does quite a few for things like ADHD and dyslexia. Mine only did screenings, not assessments, but some do.
I'm so sorry if that was all just ramble, sometimes I don't know when to stop and I just want to help. The list is great and there will be some things they should hopefully meet, normally ones that don't cost them money is the case. Feel free to tell me if what I said is useless or wrong and I can try to explain it or change anything