Hiya Tom,
NATS is interested in your medical fitness from a more long term position than the CAA. The CAA can issue you a medical which allows can form part of your ATCO licence, but they're not concerned about whether you'll pass your first renewal (or any future ones before that).
For example, restrictions on eyesight at NATS are tougher than the CAA requirements as the company worries that if your eyesight is deteriorating, you may not be able to hold a medical certificate for your entire career at NATS. Therefore, they set a tougher limit to give you some room for change as you get older.
Not sure what's happening with the future medical checks as I've already got my CAA medical certificate, but I'm sure NATS will always be interested in some way!