Good luck. FWIW, I wasn't able to find a private clinic with a decent reputation that would do anything more quickly, but that may be my bad luck. Also, once I knew it was a cyst, the private options would simply have removed the ovary, cyst and all. On the NHS several people said I'd probably lose the ovary, but in the end the consultant who operated managed to save it and it does seem to be functional. The NHS do so many more of these operations that the people who do them are sometimes very, very, very good at what they do.
I know none of that may apply to you (hopefully it's not even a cyst, and resolves!). I just had a fairly rocky road through being told I'd lose the ovary and wish I'd known more earlier on. I recently got chatting to a friend of a friend who went private and had the ovary removed, too - she was gutted. Though, she is now pregnant, and I seem to remember someone on this same thread saying they'd also got pregnant with just one ovary (that might have been kwick, actually!).