Club MSV operate all 'official' public track experiences at many UK racing circuits. However Oulton Park sublet the drift track to Driftone, which is what you saw.
Check out this site: http://www.clubmsv.com/car-home/drifting.aspx
You will be totally fine in your car, and you get free instruction too. It doesnt touch your tyres as its a special drifting circuit with a high glass content, polished tarmac.