Just to confirm that you CAN do a renewal at CAA LGW. Allow about 2 hours on the day.
You need to book at least 2 weeks in advance to get the day & time you want. A week to 10 days ahead usually has some appointments available but the choice is severely limited or not available with less than a week's notice.
You will be asked to pay 'up front' by credit card when you make the booking and you will need to take a photo style I/D ( passport or driving licence) when you turn up.
The medical is very straightforward there, you are not under any 'extra' scrutiny.
Note: I do not work for the CAA..