Just to mention if you're in the UK (I'm assuming so) you can exercise the Right To Choose to ask for a referral to an external organisation like Psychiatry UK. RTC is still "free at the point of use", you're just making the elective choice to be referred there.
(It can be easy to misread because you can also approach PUK and pay for assessment. However, that doesn't affect where you'd end up in "the queue" - just how you enter it.)
Full disclosure: I used their service, but am otherwise unaffiliated.
Unfortunately it appears they've stopped accepting new ones for now, but the information is still valid.
https://psychiatry-uk.com/right-to-choose/
(It can be easy to misread because you can also approach PUK and pay for assessment. However, that doesn't affect where you'd end up in "the queue" - just how you enter it.)
Full disclosure: I used their service, but am otherwise unaffiliated.
Unfortunately it appears they've stopped accepting new ones for now, but the information is still valid.