Just thought I'd add a few thoughts.
I'm a psychiatrist based in Oxfordshire but as a student I was under mental health services myself in SW London (Clapham I suspect comes under St George's/Springfield). Many, many people may end up seeing a psychiatrist at some point in their lives - we see small children with autism/ADHD etc., we see people with mood and anxiety issues where primary care isn't working, we see people with learning disabilities, people who have lost touch with reality (psychosis), people with addiction, and at the later stages of life we see people with memory problems and dementia. There is no "typical" psychiatric patient, and even though my current speciality is psychosis many of my patients have degrees and are employed in normal jobs.
Psychiatrists specialise in the diagnosis and treatment of mental illness, but this includes providing psychological interventions or spotting when to refer to a therapist. As part of my training I had to study CBT and took on a long psychodynamic case. So we definitely don't just prescribe!
Waiting lists vary massively across the country but are definitely increasing. The wait for my service (EIS) is under 10 days. The wait for the NHS CMHT I worked for last year was 1-6 weeks, depending on urgency. When I was a student (2005) I was seen within 4 weeks I think, and then I was seen monthly by a psychiatrist for years just because I had recurrent depression. But I do believe the many, many voices I hear who say they waited years or couldn't see a psychiatrist at all. Unfortunately whatever the government says, mental health services have had real cuts and there is a chance your daughter may not be deemed unwell enough (there now, usually, needs to be a level of risk involved before a GP will refer to secondary care). If you think your daughter is likely to only need intermittent outpatient care then finding a good private psychiatrist is an option, but be warned that if she were to ever need inpatient care this is usually unaffordable without prior health insurance. Sometimes private psychiatrists refer patients back to the NHS if the case feels too complex or risky for an individual to handle (the NHS benefits from working in larger MDTs so can provide 24/7 outpatient crisis input which the private sector can't).
I hope your daughter gets the right help soon. For what it's worth, despite benefitting from seeing many psychiatrists over the years, largely for medication input, the person who has helped the most is my psychotherapist.