Just to follow up on my last post - I have now been to the the NATS medical centre at Heathrow (underneath the tower) and can confirm that Radar is absolutely right. NATS have their own medical centre situated at Heathrow tower.
As I understand it, the doctors are definitely NATS employees, but are certified by the CAA to be able to do initial UK Class 1 medicals and of course renewals. You don't need to go to Gatwick for an initial class 1 as an ATCO with NATS.
I already had an intial UK class 1 which I got earlier in the year and the NATS doctor treated me as a renewal - although I had to do most of the medical again as it was my "initial" medical with NATS. It was explained to me that NATS itself keeps the medical info on record and it would be easier to do the medical again than NATS asking for my permission for the CAA to copy all the info and send it to them.
The doctor explained to me that the CAA only make sure that I will be fit for the next year or two until I need to renew my medical. NATS on the other hand want to make sure that I will be fit for the next 20 or 30 years as the doctor would have to recommend whether or not I would be suitable for the NATS pension scheme!
Oh, regarding the eyesight requirement - the limit is -3 /+3 at the moment for an initial UK class 1 medical as per the link I gave in an earlier post.
Hope this helps.
p.