Hi - I am currently getting private care at the Lindo Wing, which is attached to St. Mary's Hospital. My consultant is Lorin Lakasing, who has been excellent so far (I am 27 weeks). She was recommended to me by my doctor and so I contacted her directly to set up an appointment, but I believe you could also find a consultant by contacting the Lindo Wing. (If you search for Lindo Wing on the MN discussion boards, you'll come up with a number of other consultants recommended by other MNers.) I also had a recommendation for a consultant at the Portland, but they do not have full facilities in case something goes wrong and therefore would have to transfer you, so DH and I felt more comfortable with the Lindo Wing.
As other posts mentioned, it is rather expensive, and you are required to pay hefty deposits for both the consultant and the Lindo Wing itself in advance, which work out to about £7,000 total. All in, the cost will probably be about £10,000 to £12,000, maybe more. If you have private health insurance, you should check with them to see what they cover - policies differ (for example, some will not cover an elective CS; some only cover costs to a certain cap; some, such as mine, will cover the costs but not the deposits themselves, so you will be out of pocket for the deposits until after the baby arrives, etc.).
My understanding is that consultants can book up very quickly, so if you want to go private you should make an appointment as soon as possible. How far along are you? The guideline I have heard is to try to book by 10-14 weeks, but that doesn't mean that a consultant will not have space around your due date if you are already further along than that.
I'm not sure what the situation is at other private hospitals, but I am fairly certain that all rooms at both the Portland and the Lindo Wing are private.
I hope that helps - happy to give more information if you'd like!