Long post, apologies.
As others have said, speak to your GP.
Your GP already knows your case and that you need to see a specialist. Just make it a phone call, no need to waste a GP appointment. Your GP will know who is good/bad/indifferent at sorting your particular problem locally. If it’s an ENT chap you need then chances are he does private work anyway.
And TJ is mostly but not entirely correct. If you see a specialist privately then all investigations and treatment that result from that referral must occur on the private side of the fence. Without insurance it will be very expensive. If your specialist moves you directly from private consultation to NHS waiting list then he should be hauled up.
However if you do not wish to pursue private healthcare beyond the “diagnosis and recommendations for treatment” stage then that is your choice. The specialist can ask your GP to refer you on the NHS. You might get sent to another hospital entirely but GPs are not daft, they will most likely refer you to the same guy who saw you privately because he already knows you and remember, your GP thought he was the best chap in the first place.
You will then wait with everyone else to get seen by the same specialist or one of his wider team. They *could* make you have a “new patient” appointment but that would be unproductive to the point of being vindictive: it’s not the job of the NHS to punish patients for having seen someone privately. And that’s before we consider that the new patient slot you have been made to use unnecessarily would have been wasted and could have been used by another patient instead.
And since the “diagnosis and recommendation for treatment” has already been made by a trusted colleague then skipping straight to treatment at that point is the most logical and indeed fairest way to proceede.