I learnt with Skydive Ltd at Sibson and still jump there.
Chris Allen who runs it is Chairman of the BPA, a PPL and ex-Red Devil. If you want to learn they run a very good AFF course where you jump straight into freefall with two instructors either side of you. There are a series of progressive jumps and the course runs for about twelve jumps.
If you just fancy a one off then they will also organise a tandem jump were you are harnessed to an experienced jumper. They can also give you details of how to organise a charity jump.
The club at Sibson is also good for tandem jumps.
Otherwise most clubs will organise a tandem and as PPrune Towers says, a search of the BPA website will give you info on various clubs around the country.