Unless you already have a CAA class one medical for some other reason (pilot perhaps?), you will have the NATS medical first at Swanwick.
If there is something at the NATS medical that the doctor might wish to have checked further with the CAA for some reason, then you will be sent to have a CAA class one medical at Gatwick, which will be organised by NATS on your behalf. It doesn't always mean you've failed the NATS medical, it just means they want to check that you will pass the CAA one, as it's the law to have a class one medical pass to be a valid ATCO. Mostly I've heard it's people getting checked up for eyes, or dodgy outcomes one the ECG that need verifying. Nowt major. Waiting times for going to the CAA can be anything from a week to a couple of months. Depends on many factors but I haven't heard of a massive wait.
If you pass your NATS medical with no dramas, then you will not go to the CAA medical at Gatwick until sometime during the college. It is usually at the end of the Basic Course (third of the way through) if you go Area, I'm not sure about Aerodrome, but you go sometime. And again, is all organised by NATS.
Now you know, now you can chill!