I can see no way that your daughter could successfully challenge the specialist's right to levy a fee for his services. However it MIGHT be possible to challenge the actual sum involved.
As an analogy, suppose that you called a plumber to fix a leak but (unwisely) forgot to ask him about his charges. You wouldn't be able to simply say "Sorry mate, I never agreed to pay you anything". A court would rule that you'd entered into a legally binding contract to pay the plumber a REASONABLE sum for the provision of his services. (i.e. he couldn't get away with doing the job and then saying "Oh, by the way mate, I charge one million pounds per hour").
Your daughter is in a similar position in that she has engaged the services of a professional (whom she could reasonably expect to levy a charge, since that's normal practice in private medicine). He then has the right to demand reasonable payment for those services. If your daughter refused to pay all, or part, of the sum demanded from her (and the consultant took legal action against her), a court would have to decide whether the sum demanded was 'reasonable'. If the specialist could show that he'd charged the 'going rate', and that all of his other patients are charged at the same rate without any problems, the court might well rule in his favour. However if your daughter could show that the sum charged was excessive (by normal standards within private medicine) the court might rule that she should only pay the true 'reasonable' sum for such services.
Chris