cornsilk, can't help with recommendations, really, but we did look into private diagnosis at one stage, and got a list of private Paeds fom SOS:SEN. Google them and give them a ring?
Just wanted to offer you some slightly different advice, if I may. Usually, waiting lists for private diagnosis are quite long. When we were going though this earlier this year, we couldn't find anyone in London / the SE who could see us sooner than three months - which is why we didn't go down that route.
If there is ONE thing I have learned from the rocky road to diagnosis, though, is that pushing and pleading and demanding can have excellent results. I started firing off emails left right and centre when CAMHS dithered too long. I wrote to the Paed we had originally seen at the CDC, cc-ing in CAMHS, the school, everyone who had ever seen DS, saying that the situation had now reached crisis point and outlining the reasons that a quick diagnosis was crucial.
Nobody replied. But they ALL read that email, you mark ym words.
I then started ringing people up. I hammered the Paed's secretary for cancellations (I think I might have cried on the phone a few times ). I rang up the multi-disciplinary team centre where his assessment was supposed to eventually take place and made my case again to them. I got a lot of 'nothing we can do'....but low and behold, the Paed squeezed him in a week after my phonecall. He was diagnosed that day.
I also got my GP to write an urgent letter to CAMHS/the assessment centre/Paed urging them to assess sooner rather than later.
I'm not saying that putting pressure on them works in every area or for every case, but it is worth a try. I became absolutely demented - like a dog with a bone - and I think it was probably not great for my mental health (!), but I would NOT take 'no' for an answer and I became like a broken record: 'My son NEEDS a DX NOW'.
Good luck.