The medicals take place after Stage 3, and they should be considered the final stage before acceptance.
Your own doctor may be able to tell you you're fit and healthy, but NATS require you to pass the CAA medical (required to legally hold an ATCO licensce) and the more stringent NATS medical.
Basically, CAA medical covers you for the next couple of years, and as such you could pass with problems which are detected, but not serious enough to fail your medical with, at least for a few years.
NATS would like to employ folk who are going to stay healthy in their current shell, with no real problems due to pop up within your career. This is why it is a stricter medical - they want you problem free for both the CAA medical, and for your employment term.
Hope that answers your questions,
Andy