We did this with DD1 when she was coming up for 6 (she's now 8.5).
She always had extremely large tonsils and although she didn't get tonsillitus, she did get very bad ear infections. Eventually it was detected that she was suffering from quite severe hearing loss in one ear. 2 GPs and School Doctor spent most of their time referring back and forth to one another. Another problem was that she used to stop breathing during the night - only for seconds at a time, but she would keep jolting and waking herself up. This in turn made her extremely tired during the day. Her everyday life was starting to suffer - very tired, couldn't hear properly and STILL they refused to refer us to ENT.
Went to a different GP (same surgery) who was in agreement that she should be referred. We said we would pay to see the Consultant. We did this and was seen the next week. He was absolutely horrified that we hadn't been referred to see him earlier!! He told us that she had "severe respiratory obstruction" - he wasn't concerned about the hearing/large tonsils as such - just the fact that she was suffering breathing difficulties at night.
He offered to put us on the NHS waiting list, but we said we would pay. She was taken into hospital 4 weeks later and the operation in total (including an overnight stay for both of us - and all my meals etc) was £1640. At that time, we didn't have Insurance, so we paid ourselves.
I have absolutely no regrets at all - her life has improved 100%. Hearing is back completely, her breathing at night is normal (spent first week afterwards sleeping in her room). She has far more energy and looks a lot healthier.
DH had exact same problems last year and had his removed privately - although he had his done on the Health Insurance provided through work.
DD2 (4.5) is showing same signs as DD1 did at that age - stopping breathing etc so we may end up going down the same road again.