What is causing the problem to the PCT, just the referral or are they worried you might request some sort of treatment as well?

I am surprised they are wrangling over a consultation, even for something specialised like NF I wouldn't have thought it was a problem, I work for a PCT and I've never heard of anything like that.

Although as the others say, if you are happy to pay and have the resources, going private will allow the PCT to spend the money on something else, just make sure you inform them of this, so they don't book you a referral which you won't need!