The NATS medical is to satisfy them that you're fit and healthy to work for them (and that you pass the drugs test). However, to be an operational ATCO, you need a medical certificate from the CAA which forms part of your student, and later full, licence. NATS want to check you fulfil the CAA requirements before you go for the 'proper' medical.
Initial CAA medical is issued at their offices near Gatwick Airport, after that, you can get a renewal just about wherever you are.
Hope that helps.