I'm assuming that because you've got depression and are seeing an NHS shrink that you're seeing someone from CAMHS? As far as I'm aware -- I personally haven't seen anyone from CAMHS about being trans because the only one who will see me is a sadistic freak and there's no way I'm going back to see him -- if you spoke to your GP about being trans then they'll refer you to CAMHS anyway, who will check for any other mental health issues that could lead to you believing that you're trans, and then once you get the all clear from them they refer you to a Gender Identity Clinic (GIC). There's a couple of those around the country, about half will see people your age I think, although they do prefer it if you have parental consent under the age of 18. You won't get put on T until you're 18 as far as I'm aware, but it will get the ball rolling and you can get a start on doing real life experience when you're ready so things might get sped up when you do reach 18. That would be my plan, but as I said, there's
no way I'm putting my self at the mercy of my CAMHS doctor again.
They have to keep things confidential, especially if you explicitly ask them to. Obviously if there's a risk to harming you or someone else that's different, but there should be none of that in this case, (unless you're planning to do your own top surgery

).
As for coming out to your parents, I did it through my parents closeted lesbian friend. She told them that I wasn't happy being a girl, and they took me to my GP about it a couple of times -- which just dissolved into ranting about my shrink and begging to make him close my file because seeing him was making everything worse -- but they've not spoken to me about it since. That was probably over a year and a half ago, so I think they're just hoping it does away. It's not going away. It wasn't easy, but their reactions are either going to go one of three ways, accepting, ignoring, or hating. If they you that you're "gay" that might help you gauge which of those their potential reactions might be.
What I found helps with no one in real life knowing is being myself online. One of my friends knew, but she never really understood it. I told another person that I was my ex's boyfriend, and told her that they called me "Alex" but I don't think she ever put two and two together. It's tough no one knowing who you really are, but sometimes it can still feel great, just today my friends at college agreed that I'm "practically a man" and they weren't just talking about my style. It's good to know that other people can see me through this stupid body.