That's so disappointing for you after they lost the referral. A similar thing happened to me. I was lucky and asked to be referred to Psychiatry UK via RTC and their wait times then were 4 weeks. That was 2 years ago. It's changed a bit since then, unfortunately.
I've done a quick search and the timescales for some are now very long. Psychiatry UK won't even commit to a timescale any more.
Assuming you're in England, there's always a lot of new providers coming on board with Right to Choose, and the wait times shift a lot, so it's worth searching around.
Here's one that may be woth looking at, although they still don't state a timescale. Problem shared do Autism and ADHD assessments.
https://www.problemshared.net/resources/articles/what-care-does-problemshared-provide-through-right-to-choose
I've always found it a nightmare for Autism assessment providers, as you have to Google and go through every one that comes up.
For anyone thinking of ADHD assessment, ADHD uk keep a list of current providers and an estimation of wait times which is really helpful. (They also state which ones deal with titration)
https://adhduk.co.uk/right-to-choose/