In public the operation will be performed by either Dr Wainwright or one of his team. There is good evidence that the level of surgeon (overall) does not make a significant difference to surgery where the surgeon performing the procedure is "competent". Your surgery would only be performed by someone capable of safely performing it. Where possible, patient wishes are taken into account but if another surgeon has a much shorter waiting list then you may well be offered surgery by them to cut down on your waiting time.
In private, the surgery is always done by Dr Wainwright. He often works with an assistant who helps during the procedure.