UK here. Your GP is right about the massive backlog in mental health referrals (2+ years in my area) but wrong about prescription charges & the NHS recognising a diagnosis from a private psychiatrist.

Because of that huge backlog I went for a private diagnosis & prescription and then had that prescription transferred to my local GP under what's known as a 'Shared Care Agreement', so my scrip cost ~£110 per year (buy a Prescription Prepayment Certificate or 'season pass').

I know people who've managed to circumvent the waiting list (and private fees) using 'Right to choose' but I understand that can vary depending on your local health authority. Certainly worth exploring though.