Stressyhead, I worked in NHS dentistry for many years. If you are in pain and needing treatment, you will be covered under the NHS. You do not have to see anybody privately.
Have you had an x-ray taken to see if the roots of the root filled tooth are infected at all by an abscess? This would cause you pain.
It is possible to have the root filling removed and re-done. Sometimes there can be 4 root canals and the dentist only finds and cleans 3 so this can still result in on-going pain.
Can I ask is the pain worse if you put your head by your knees (standing up)? This would tend to suggest a sinus problem.
If the pain is chronic and been going on for so long why not ask your GP to refer you to an ear, nose and throat specialist?
In my experience dentists used neuralgia when they didn't know what else could be wrong!
HTH