I've heard of people getting referrals to private consultants for an initial assessment, who then put them onto their NHS list. It seems mostly to be people who urgently need joint replacement surgery, though. Where they have such severe pain that their employment's at risk, consultants doing both private and NHS work seem able to hurry things up quite significantly.
Doubt it would work like that for ME/CFS, though, even if we had consultants. A badly knackered knee is a problem that can often be solved or improved to 'good enough' in one shot, so people look at it differently.