Therapy in London splits between the NHS and private practice, and the two offer different things.
NHS Talking Therapies takes self-referrals. You don't need to see a GP first. You refer yourself directly, usually through an online form, and you need to be registered with a GP in England but not to have an appointment or a referral letter. Nationally, most people start treatment within six weeks, and the median service begins treatment about three weeks after referral. It's free.
NHS Talking Therapies treats anxiety disorders and depression, as a defined course of sessions, usually cognitive behavioural therapy or another of the approaches recommended by NICE. It doesn't offer a choice of therapist, a choice of approach, or open-ended work.
So the case for looking privately in London is choice: you pick the person and the way they work, with no referral, no assessment to pass and no fixed endpoint. There's nothing to claim back either, since private therapy is paid directly to the therapist, though private medical insurance sometimes covers it, usually requiring a GP referral and a therapist on the insurer's own list.
What to look for, because the titles will not tell you
Psychotherapist, counsellor, therapist and psychologist are not protected titles in the UK. Anyone can use them, with no qualification and no professional body required. Only nine practitioner psychologist titles are protected, including clinical psychologist and counselling psychologist, and those require registration with the Health and Care Professions Council (HCPC). Psychiatrists are doctors, regulated by the General Medical Council.
So the law asks nothing of a UK psychotherapist or counsellor. We ask for more. A psychotherapist listed on It's Complicated needs a master's-level qualification in psychology, counselling or psychotherapy, and can only name a specialisation if it was a main focus of their training. A counsellor needs at least 500 hours of documented training and registration with a recognised professional association such as BACP or UKCP. Clinical and counselling psychologists must hold HCPC registration, and psychiatrists GMC registration.