For implant procedures carried out by a private dentist, will I have to pay for my factor treatment?

Yes, you will have to pay. Implants are unavailable on the NHS, so it is prohibited to have NHS and private treatment concurrently for the same condition, regardless of what it is. It doesn’t only apply to dentistry. It’s unfortunate that implants are not included as the accepted standard of care in the NHS as they are so expensive and simply have insufficient resources in the NHS to justify high-cost treatment when simpler options for tooth replacement are available. Implants are available on the NHS for patients who have significant loss of teeth/function through cancer surgery and significant trauma.