Hi pseu,
I'm a medic and did a psych Bsc and an Msc in clinical neuroscience previously at UCL.
Psychology really will not satisfy your interest in neuroscience or mental health; < 25% of a psych degree would comprise those (and even then they'd be pretty dumbed down biologically speaking). There are lots of tailored masters degrees in mental health and neuroscience, but sadly you would need a biological based degree to get on those so they're out of the question.
Your best bet does sound like a neuroscience degree with a strong clinical component. Psychiatric neuroscience will be covered as part of clinical neuroscience not cognitive neuroscience, but even so, there will be general bits of neuroscience and molecular biology that may not appeal to you. Also, almost all neuroscience degrees require either chemistry or biology A level. I'm assuming you took art A levels? In which case it may be hard to be accepted on to any neuro undergrad courses.
But to be honest as punctuation says it seems like a big investment for just an interest. Instead of investing £40,000 studying, you could just buy a few books, go on a few short courses, read a few blogs and chat to some students.
Psychiatry and neuroscience are my areas of expertise, if you have any burning questions feel free to fire them my way.