The NATS medical is a company medical and also drugs test. It is practically the same as the medical which you will also need to have at the CAA at Gatwick, if not a little more rigorous.
Once the NATS medical at Swanwick is passed (you will know on the day), then they will send you to the CAA at Gatwick. This is required to issue your initial UK ATCO Class 1 medical certificate after which NATS will renew it for you annually.
Even if you have a valid JAA Class 1 medical (for commercial flying licences) you will still have to go to Gatwick for the ATCO medical (although it's practically the same examination, some tests may be within validity from your JAA Class one - e.g. audiogram, ECG, spirometry etc.). In the good old days NATS had an arrangement with the CAA to issue the initial themselves, but sadly this is no longer the case (since the previous head of NATS medical department left after the move to Swanwick).