These people use circuits all over the country. Do 3 cars minimum though, you are just getting used to the speed and flappy paddles by the time you get out of the 3rd car.
http://www.everymanracing.co.uk/junior-experiences-emr
My wife bought me a 3 car experience for my birthday with them and it was fine – yes, lots of hanging around, not a huge amount of track time and they will try and sell you many extras! Be prepared to pay double the original price by the time you’ve done.
All 3 cars I drive were new and clean and I drove way faster than I’ve done before and the instructor didn’t try and slow me down.
And if you can stretch to it add a couple more cars on the day – they were selling additional rides at discounted prices when I went.
This company get very mixed reviews though, but most of the negative reviews seem to be from people who aren’t allowed to pretend they are Lewis Hamilton. They are just an experience, not full tuition, race training and getting the back-end out session!