I would ask your current doc to recommend someone, if her condition is really that rare they will know people in the UK. If it isn’t so rare the international community all know each other then I’d expect any clinician to be able to manage her.
UK healthcare isn’t really set up so that the “best” doctors are professors. In addition those who are often devote their non NHS time to research so don’t do much, if any, private work.
Virtually every private doctor will have an NHS practice and if your child’s condition needs input other than from a doctor you might be better to look at NHS care where the team based structure is much stronger and allied health professionals routinely work together eg. Speech and language, audiology, specialist nurses for family liaison,
This can require quite a mindset shift if you are coming from a different system.
As others have said the two biggest NHS providers for specialist ENT in London are GOSH and Evelina. The teams in both places are strong and many surgeons have private practices. Names that came to mind especially for ears are Mark Felton, David Albert, Dan Jiang & Ben Hartley.
I’m pretty sure that all of the people I’ve named work in the private sector but not everyone good does and not everyone that works privately is good. If you decide to use the NHS then don’t discount someone just because they don’t work privately...i’m not an ENT surgeon but amongst my colleagues