I'm not sure exactly about Scotland, I'm in England, but seeing as it's still the nhs I would imagine the process is similar.
Basically your doctor needs to refer you to a GIC. To do this you have to go to a one off, hour long session with a psychiatrist who will then either support your referral or recommend a different course of action if he/she thinks there's underlying issues. Depression is not one of those issues if it's caused by your gender dysphoria, so don't worry about it, it's fairly straight forward.
After that your gp will take blood tests and probably request an examination of your genitals, but you don't have to do that if you don't want. Then they can write a letter with the psychiatrists supporting letter to the GIC.
After that, in England at least, there's a fair wait before anything happens. You'll most likely get a letter saying you're on a waiting list. Once you finally get an appointment you'll have an assessment period of at least two months but potentially much longer depending on what stage of your transition you are at. If you have transitioned socially and at work and are doing ok then the assessment period will be shorter, but if you are at the stage where you are still in the closet and trying to figure out whether you want to transition or not then it will take a lot more appointments. At the end of the assessment period they will either make a diagnosis of 'transsexualism' or send you back to your gp.
If you get the diagnosis, you are then on the care pathway, and they'll work out a treatment plan with you. Again this all depends on your circumstances. They won't prescribe hormones until you've been living full time as your preferred gender for at least 6 months. Surgery comes after 2 years full time.
I was referred in June last year, and had my first appointment at the GIC in June this year. I went full time last November and started hormones privately in February. This meant my assessment period was only 2 sessions.
It is slow, and after that first referral you are basically left on your own for a bit, but you can always ask your doc if there is any kind of counselling available, even if it's not gender specific. Once I actually got to the GIC (Leeds in my case) the people there have been great and not the evil gatekeepers is been warned about. Some people just have unrealistic expectations, they expect to be able to go along as a guy who isn't out to anyone and get hormones, then get angry when it doesn't happen.
Hope this helps, like I say it might be a bit different in Scotland.
