The short answer is that the Law has nothing to say about it.
Even if there was a charge for training - that wouldnt mean that you could muscle in on the deal and say 'and that means you have to treat me.
Ther have been a series of cases about forcing either doctors to treat or Hospitals to admit and treat and basically the judges have shied away from ordering treatment
Yes you have a legal right to a dentist, and you are correctly concluding that therefore the secty of state has a duty to supply care.
actually that is s1. of the 1948 Act, and I read that in 1973 and suggested to the other law students that on the back of Blackburn v MPC 1968 the judge might grant an order to this effect.....It was Hohfeld who first said that if A has a right from B then B has a duty toward A.
The first case was Hinckley 1976 - and the judge held that a minister can owe a duty to a group of people ( that would be you patients ) and yet not owe a duty to one particular person.
There were a few other tries - one over emergency treatment ( Collier ) and I think that was the one where the judge walked in at the start of the case and commented: Don't we know the law on this ?
adn the barrister for the applicant tried to persuade a doubting judge that this time it was an emergency, and so the principles in Hinckley dont apply blah blah..... the judge held that they did ( hold ) - order not granted.
So you could start suing but it may be expensive - legal aid would not be granted as the principles of law are clear.....