You certainly can 'jump' NHS waiting times.
Dd has complex medical needs. At the time we went private for consultations she was NG fed but pulling the tube constantly so it was reinserted for every feed ( under one so lots of feeds) I had to reinsert it myself or go to a and e every single time.
The NHS wouldn't do gastro surgery for a more permanent tube until we had a diagnosis. Appointments with various departments took months at a time.
Once we had a feeling what was wrong we asked our ENT to check for this. They wouldn't and their reasoning was it was very rare so wouldn't be that so was a waste of a general anaesthetic etc.
We went private and spent around 5k for the consultations and the procedure to check. ( our hunch was correct and she did have what we suspected)
The surgery to fix the problem and to sort her feeding tube would have been around 20k.
Our consultant bounced us back to himself to his NHS clinic and we had the procedure done within 2 months. He arranged another surgeon to insert a PEG at the same time as the NG would damage the throat work he was going to be doing.
The NHS waiting list at the time for her PEG was close to a year. We'd still be waiting now to even have a diagnosis... We spent a year in the hands of the NHS with a baby who was clearly poorly and it took months to even get someone to listen and once they did start to listen we spent a long time searching for answers.
I do feel guilty that there's other mothers out there who can't afford this but we did what we did as we needed answers so I don't feel guilt on our part exactly - if that makes sense?
If we had to we would have funded the 20k with loans but the specialist refered us without us asking him to.
Her surgery has totally improved her quality of life and I'd do it again if I needed to.