This may be too late - sorry if it its;
If you've had 2 unsuccessful courses of antibiotics with your GP then they can refer you. It does take a while, but often you'll end up seeing a Dermatologist who does both NHS and private work, so it's really about whether you're happy to wait. I saw one privately last year (Dr Tamara Griffiths in Manchester if you happen to be in the North West!) and she was great, though mine is *slowly* getting worse (and I'm on my 3rd antibiotic) so I'm going to request a referral to a dermatologist on the NHS.
Bear in mind that, whilst your GP might not know as much as dermatologist (and I know how frustrating this is!), they're able to prescribe ANYTHING (bar oral Isotretinoin - (Ro)Accutane) that a dermatologist could - so if you do your research and request a certain topical/systemic treatment then they will probably give it to you.
Be wary of going to a 'clinic' - whilst they offer fantastic treatment and service, they often sell 'procedures' as opposed to medical treatment, so I'd ask friends/medical professionals for names if you do decide to go private. I'd also look at the dermatology staff at your local teaching hospital - many consultants also do private work and if so, there's likely to be a fair amount of info about them online!
I hope this helps; I know what a pain GPs can be with acne!