A consultant cardiologist is a consultant cardiologist. Accreditation is the process by which a doctor becomes a consultant, and they dont specialise in physiology! The problem is that the CAA will want the report to confirm specific elements which are often taken for granted by doctors and so not written down in non CAA reports, so I agree the best thing is to ask your AME for the name of a cardiologist who has done this before
You will probably need to see them privately in a private hospital. The cardiologist will normally work in the private hospital nearest their NHS hospital simply for convenience. This may be run by Circle or Spire, but they are just hospital operators, dont 'employ' or 'own' consultants. The doctor is just renting space and so the name of the hospital/landlord is totally irrelevant. In London cardiologists work in rooms and this may be cheaper and faster, but always get a quote in advance.
And finally, although opinions can differ, for a standard test such as an exercise ECG it matters not a jot. I am sure you have enough to worry about without making a meal out of a simple task of finding a doctor!