Within the UK? It's likely to depend, but I'm pretty sure it's nowhere near the rates their US counterparts make. This may well change with the new plans to subcontract NHS services among private firms, if that manages to get off the ground :s
It doesn't really matter though, as you'll be making more than enough money at the rate of an NHS consultant pay to live very comfortably, regardless of whether you work privately or in the NHS.
The astronomical rates the US private doctors (especially surgeons) make really just means the difference between "nice house, car, clothes etc" and "several nice houses, cars, and so on" which is kind of irrelevant ultimately as you can only reside in one house at a time and drive one car at a time...
This is beside the point that the oft cited salaries are considerably lower in real terms due to the very high cost of malpractice insurance etc in the US - and the overhead costs in the US of running ones own practice increasing quite a lot these days (and the private health companies that exist there, in the same vein as BUPA or virginhealtcare, tend to pay a lot less than figures would suggest from what I hear).