I don't know where you live, but if you want a prompt diagnosis by a psych of your choice, then I would choose to go private. I live in Canada, and I totally believe in free health care. Without going into too much a tangent, I will also add that I do believe all psychologists should be covered under the universal health care because our mental, emotional, etc health should be just as important as our physical health--maybe even moreso sometimes. Anyway, because this is not the case here, I did go private. This was to avoid the year or two long wait lists, and also because I knew exactly who I was getting and whether or not they were good quality. If I had to do it over again, I would do it the exact same way. If you don't have the money though (which I didn't, but I was willing to go into debt for it) then you might not have a choice but to try and get on a wait list in a public clinic (this is assuming you are in Canada or somewhere else that actually offers such services).
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Diagnosed with classic Autism
AQ score= 48
PDD assessment score= 170 (severe PDD)
EQ=8 SQ=93 (Extreme Systemizer)
Alexithymia Quiz=164/185 (high)