Hi again

. My experience of the process was this (and stage 2 is no longer necessary, since April last year; if the gender clinic has its own psychotherapists or psychologists, your GP can refer you direct to the clinic - so it's good to call the nearest clinic and ask what's required, unless this is already rolling for you):
1. June: asked my new GP to be referred to gender clinic - very straightforward, it's not their business to decide whether you're "eligible", that's what the gender clinic is for - so they should just refer you
2. July: referred to a consultant psychiatrist for assessment before referral to gender clinic - also very straightforward, they just want to make sure that if you have any mental health iissues going on, you're supported for them whilst you're transitioning (and to check whether your gender dysphoria is not a manifestation of a mental illness, which it very occasionally can be)
3. Sept: first appointment with the gender clinic, 3x assessments with a psychotherapist, every six weeks, to get my story
4. Jan: first appointment with clinic doctor
Typically, the clinic doc will want to see you twice before starting you on hormones - this means it's roughly 6 months from first time through their door to going on hormones (my story was different, because I was already on them by the time I showed up there, but otherwise the timeline is pretty standard).
Generally speaking, so long as you're not going to the Charing X clinic in London (who have a reputation for playing gatekeeper games), the gender clinic sees their role as that of enabling you to be your true self. And they have a duty of care towards you (as well as being accountable to their funding body), so they want to ensure that if you're going on hormones, it really is the right thing for you. Hence the wait. I was not put under any pressure to conform to any particular clothing stereotypes (I made it clear from the beginning that I'm female, but I'm not feminine). Our clinic is slowly but gracefully getting its head around some of us being non-binary in gender.
You will find people who have had bad experiences too, and no doubt some will have things to say here - it's good to hear a mixed bag. But I wanted to make sure you heard a good tale too.
ETA: This is all 2½ years ago, for me - I'm having surgery this May. The clinics and system in general is improving all the time. I hope you get to hear from some Scottish folk too.