Birthmark removal through private health care

My daughter was born with a ulcerated hamangioma birthmark on her face. We were told by the NHS that is a lot of cases they would perform surgery at the age of 4 and half. We attended the hospital for her birthmark removal to be told she had been placed on another 1-2 years waiting list due to the lack of pedestrians for cosmetic surgery. I have found out that I can get the same treatment much sooner on private health care in Surrey and need your help. My daughter is being teased at school for the (lump) as they call it. As a mother I cannot let her go through this for another 2 years. Especially when it can be taken away.  Thank you