I always wondered, why is NHS dental care treated differently from NHS medical care financially? I mean it's all the NHS right? At what point does a dental issue become a medical issue, ie a serious teeth gum problem maybe turns into an infection that doesn't just affect your mouth - does your treatment then become free?
Do the British just not consider dental care as important as the care of the rest of the body? Maybe that old chestnut about the British having terrible teeth has some kernel of truth to it? 
I don't know for sure (maybe someone else here does), but I would hazard a guess that maybe any work that can be and is done by a dentist comes under the dental treatment costs, and anything that needs referral to a doctor or hospital comes under 'free' NHS treatment.
The £80 is to see a PRIVATE dentist. If I HAD the money for private care, why in HELL would I be dealing with the NHS? We're not all well-to-do wives of executives here, you know.
As AnneR suggested, if you aren't willing to pay £80 to see the private dentist, there's no harm in trying to find another dentist who can do it on the NHS for you. However, keep in mind that whether you go private or go with the NHS, you will still have to pay for your treatment - the only difference will be the amount you have to pay. It could be that your complete course of treatment on the NHS will come under the £198 band, in which case you would have to fork out almost £200 for the NHS services anyway. It might be worth contacting the private dentist to find out how much they would charge for your treatment and then you can weigh up your options.
At the end of the day, you need to get the treatment, but you are also going to have to pay for the treatment, whether it's NHS or private - so it's up to you which course of action to take: a) wait the 5 months (obviously not ideal, especially if you're in pain), b) find another NHS dentist who can do it for you now, c) go to the private dentist.