I don't think you need a psychiatrist to get diagnosed with anything. A psychologist is usually the one who does the diagnosing. Psychiatrists are more for prescribing medications. Unless the whole system works differently in the UK?
I have similar trouble trying to get psychiatric care (my insurance doesn't pay for it at all and there are precious few English-speaking doctors in this country, most of whom are just awful), so I feel your pain. If you're not satisfied with your current therapist, I encourage you to take steps to find a new one, as that makes a HUGE difference. There are so many different schools of therapy, and none of them works for everyone. If you think your current medication isn't helping, it's true that you should give it a little time, as SSRIs take a long time to start working. But it's also true that no medication works for most people, so if you think you've waited long enough and you're not seeing results, it's perfectly reasonable to want to try something different.
I know what you mean about self-diagnosis, too. Doctors, especially older doctors, really hate it when their patient knows what they're talking about. They see too many people who found the name of a condition online and ran to the doctor even though they clearly don't have that problem, so they tend to immediately dismiss anything we suggest. They have to come up with the diagnosis on their own, or they won't help you. I once had a psychiatrist give me medication that could have killed me because he didn't believe what I told him. However, you deserve to have a doctor who BELIEVES you and respects your intelligence. If your current doctor is refusing to consider anything you suggest, then it may indeed be time to find a new one. They're not ALL like that. At the very least, a doctor should listen to everything you have to say and at least consider it as a possibility, looking for proof that you are wrong before dismissing your self-diagnosis. You know what's going on in your head a lot better than they do. It may be that you're mistaken about what's wrong, but the doctor needs to explain why in a respectful way, not just dismiss your suggestion outright.
You can try a trick I sometimes use to soften the "self-diagnosis": tell them you have a friend with [the problem you think you have], and when they describe their problem, it all sounds exactly like what you're going through now. Do you think I might have the same problem, doc? (I'm very uncomfortable lying, but sometimes it's the only way to get the doctor to seriously consider my suggestion.)