I think that this is a difficult way to get help for something like depression because they are geared up to deal with emergencies. There are long waiting times and my experience is that staff can sometimes be less than sympathetic even with an actual injury (self inflicted). It is often uncomfortable (e.g. sitting in hard plastic chairs near the open door, lots of people trailing in with upsetting injuiries or in a very bad state) and even when you finally get seen, staff do not spend much time on your case because they have to get you through quickly (i.e. admit you or send you home) to keep things moving. However, this should not put you off going to A&E if you do need immediate medical attention for your self harm injuries.

You mentioned going to the doctor and this is the more usual route to get help for depression. At your GPs you can make an appointment, so you won't be waiting for ages and you will have your GP's full attention. You can request a double appointment if you think it will take time to explain everything. Your GP will be able to prescribe medication if you agree together that this is suitable. (S)he can also refer you to a wide range of services which specialise in treating depression. Many GP surgeries have a counsellor based in the surgery who can offer cognitive behaviour therapy.

If you want help in the moment when you are feeling bad, phoning a helpline may be more helpful to you than going to A&E. Saneline is well respected. HopeLineUK is another one. There are helplines which address mental health problems generally, and ones more specifically geared to self harm issues. There are also more local ones- have a google and see what comes up.

Mind is a mental health charity which has local centres with activities and therapies which may be helpful too. They are also good at signposting you to other good services.

You are right in thinking that because you are 18, medical professionals cannot contact your parents without your consent. However, in cases of emergency (e.g. admission to A&E) and if they think that you are a danger to yourself, your confidentiality may be broken.

I hope some of this is helpful. Happy to tell you anything else which may be useful.