CJayne:
If NATS are totally happy with you after your NATS medical, they might not send you straight off to the CAA, but at the moment, it seems that's happening anyway for most of us.
If there's anything NATS isn't 100% of, you'll definitely be visiting the SRG at Aviation House, Gatwick before you start at the college so that they're certain that you're ok and will get a Class 1 (ATCO) without any problems.
Once I'd had my NATS medical I got an email from HR a couple of weeks later telling me to book an appointment at Gatwick, and once you've had it, you just send them a copy of your Class 1.