Hello,
I am not sure if NATS's policy has changed, however everyone on the most recent intake at the college had been and passed the CAA medical as well as the NATS one.
After taking the NATS medical you will be instructed to book your medical at Gatwick, when I did mine the waiting time was around about a month.
You will need a tray in the CAA canteen as the food is hot.... don't forget to form a queue!