I look at the cornea/external disease/anterior segment consultant team at the regional teaching/university hospital in the NHS website 'find a hospital' (hospital websites are not always up to date so don't hesitate to phone the dept for info on who's who and when). You can get GP to refer to chosen consultant direct on the NHS meanwhile (GPs have always been clueless for me on who does what in hospitals and relieved when I've done the groundwork and know where to go).
Then phone the consultant's NHS secretary for the private practice number, if it's not on the NHS website. Private hospital chain info numbers have always been 100% useless and wrong for me as to who does what but you might be luckier. Basically, the dude's private secretary organises his/her patient list and tells you the cost straight up.
Maybe, if into this, do a search on PubMed to check out research interests or what they've got funding for, ie building a team of specialist registrars actually working the clinic so you get informed people covering the NHS clinic rather than generalists.
Paediatric ocular rosacea ~ primum non nocere