It might differ, depends on where you are from, but I'll tell you how it works in Canada
For autism you need a specialized psychiatrist. The easiest solution would be go to your family doctor, say "I want to get autism spectrum disorder diagnosis" so they can refer you to a clinic where it's done. You can also ask your psychiatrist you are going to in December so they can tell you if it would make sense for you to get diagnosed (like. They can not diagnose you with autism, but they can tell you if some of your symptoms fall under ASD)
For HSD or Hypermobility or HEDS you need to go to physiotherapist. Psychiatrists can not diagnose you with that, so you'll need to go to family doctor first again, they will be able to check with base hypermobility criteria (if you want to also check for HEDS, your physician should be able to do it with base criteria, but most likely will refer you to another doctor). But again, primary care physicians usually check only shit you already checked yourself and then being like "yeah go to XYZ specialist for proper diagnosis" (which I do believe they are mostly needed to justify to your insurance your visit should be covered? Dunno. I go to mine for free since clinic accepts my insurance and there are no crazy lines there, so I just was going there first to make sure my dumbass didn't misunderstood the criteria as other doctors cost pretty penny

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Oh right. Going to primary care physician first also important so they can rule out other explanations for your condition. Like they might take your blood work and etc