Hi, I am one of many, many private counsellors and therapists in this area working with people with depression. Many of my clients choose not to go through their GP, although I would always recommend keeping them in the loop, even if you choose to seek help in the private sector.
If you want a private therapist, a good place to start is through one of the main professional bodies. The BACP has a website
www.itsgoodtotalk.org.uk that lets you search based on location and issues. I think the UKCP has a similar one too.
Any qualified Counsellor or Psychotherapist should be able to help with depression but approaches vary so you might want to try one or two before deciding who to go with. Therapy is most effective when you click with your therapist so just keep trying until you find someone you feel comfortable with.
Oh and if you don't feel comfortable talking to your GP, you might want to shop around for a different one of those too!
Best of luck and feel free to PM me if you have any specific questions.