I learnt with The AA and I liked my instructor so much (easy to get on with, good teaching style, etc) that when he quit The AA and set-up an independent I switched over. It even worked out a few bob cheaper than The AA.
The biggest thing is the car you learn in. BSM are switching to Fiat 500's -though currently still using Corsa's, the switch of vehicle will hinder you somewhat if it happens the day of your test- so expect them to spend their life in the garage getting fixed, The AA use Ford Focus's, which comically the company they lease from used to have RAC break down cover, but I digress, there are a few independents round here that use poverty spec BMW 3-Series. All of which will make learning to drive somewhat more interesting.
The best thing to do is ask around your friends and family who have passed as to who they used and what they thought of them. But ultimately, go where you think you will learn more and remember that statistics mean nothing to the individual. BSM have a large number of new students each year so their average is going to be lower. All it does is highlight the range of driving ability students have who join that driving school.