In the UK the family doctor (GP) acts as a gatekeeper to other services. Thus, seeing your doctor is excellent advice for people in the UK who are depressed. Your general point - don't get the treatment from a regular doctor - is a good point.
In theory you should get a quick assessment, and then a referral to PCAT team (Primary Care Assessment and Treatment) - these people will either take you on for short form talking therapy (probably cognitive behaviour therapy) with possibly some medication; or they'll refer you on to a community team; or they'll send you back to the GP.
The doctor may skip the PCAT referral and send you straight to a community team. They'll do an assessment and either take you on, or send you back to the GP with advice for treatment.
If a community team take you on you'll get a care coordinator, a care plan, a consultant (psychiatrist), and access to different meds and different therapies. Frustratingly, you might not be allowed access to short form CBT while registered with a community team. If you need hospital you'll get hospital.
In an emergency the GP will just call the crisis team for you (in theory crisis teams are available to anyone who needs them at any time, but it's usually better to go though a GP unless the crisis team already know you). The crisis team will assess you for a stay in hospital (either informally (voluntarily) or under section of the MHA (detained against your will to protect your safety)).
Of course, that's the theory and often the practice differs.
And, you can avoid some of that process by getting a private psychotherapist. Look for BACP registration. (There are probably other good registration, but BACP is the one I know.)
About psychiatrist:
I've met lots of psychiatrists.
They've almost always been "odd".
To become a psychiatrist in the UK you need to do the regular medical school. That turns you from a medical student into a doctor. (But not a doctor that can practice unsupervised.) You then do Foundation Year One and FY Two. (This is practice, supervised, in hospitals etc.) You then become a registrar. Eventually you become a consultant in your chosen speciality, which gives you the power to practice unsupervised.
All of this means that you have great, up to date, in depth knowledge of modern treatments and medications. It also means you're a bit of a basket case to make it through that process. Your bedside manner is likely to be interesting.
I'd strongly recommend people avoid psychiatrists unless the illness is severe or resistant to other forms of treatment.
As always: I am not a doctor and I have no training and all of this is just my opinion and etc etc.