Mental health is tricky. It's a very long story actually that dates back to deinstitutionalisation in the 1960s, and isn't just about payment. The system now emphasises inpatient as only a short term stabilisation measure. In fact, medicaid is barred from paying for long-term treatment in mental hospitals over a certain size because when it was established in 1965 they didn't want it to subsidise the mental hospitals that they were trying to close. It was sort of a good theory but they never built the comprehensive outpatient system that was meant to replace it (you can thank Reagan for that) and it's become clearer since then that for patients with serious mental illness you need a comprehensive wraparound strategy or they bounce in and out of treatment, sometimes becoming homeless.
The ACA requires mental health parity, but the system for valuing services and determining payment is flawed, so providers are not always paid enough -- doubly so for patients on Medicaid, which the sickest are. And there's a perennial shortage of providers, especially in rural areas. Because you cannot force doctors and therapists to participate in insurance, some providers, especially in large urban areas, do not accept it and you have to pay out of pocket; depending on your coverage you might get reimbursed in part.
However, given the horror stories I am hearing about mental health care on the NHS these days, especially CAMHS and autism services, I am not sure that I would choose either the UK or US for mental health care if I had an option.
I will say that the US is generally very good at provision for younger children, particularly children with autism and developmental disabilities. Many states have medicaid programmes that cover children with disabilities independently of their parents' income (meaning all treatment and therapy is free). All states have an Early Intervention birth-to-three programme that provides services in people's homes (speech, physio, OT, special instruction as needed) and special education preschool services. 2 of my 3 are autistic and have been seen at one of the best children's hospitals in the country and the care has been superb. I paid out of pocket once for a specialized evaluation (neuropsychological assessment) but that was to bypass a wait list and I got part of the fee back. The only thing that wouldn't have been covered is if I wanted a psychoeducational evaluation as that isn't considered medical. (Usually the school does this, but sometimes parents get it done privately for various reasons, usually because the school fights them about doing it.)
Care for adolescents with serious mental health issues is harder; there's some evidence that too many inpatient beds have been cut for them.
I would not say all my experiences with the US health system were wonderful and I wouldn't defend its structure (despite working in it!) but I will say that both the health and special education systems have been excellent for us.