I have personal experience of three schools in the North of UK and all three were CAA Approved Schools as was
I agree with the information about Multiflight in leeds although strictly speaking the weather doesnt actually slow the CPL course down at Leeds much more than other schools in the area, it does however suffer from quite sporting cross winds and is a tad higher than most aerodromes but overall the CPL is a bit of a lottery Wx wise wherever you do it in the UK. But I would recommend Multiflight based on my own experience
I found LFS okay but I understand they have recently been taken over, it may pay to speak to students there since the take over, I know one of the instructors is still there and he was a good enough sort
I did do a lesser amount of training with them but had to break off due to personal reasons but found them to have a similar approach to multiflight
Both Multiflight and LFS were well organised and I couldnt fault the quality of instruction and it would be hard for me to to pick one above the other but if pushed would personally recomend Multiflight simply due to the fact I completed a much larger section of my training with them
The third school based at Humberside were in my opinion poor overall
My advice would be to go to Multiflight if they can fit you in, if not then try Leeds Flying School, if they are full wait until you can get in at Multiflight
Better still speak to recent trainees of all three if you can manage it. Any feedback from an individual will reflect their own experience my advice would be to get the view of as many people as possible, as much feedback as possible and visit each of your possible schools before making a decision.
Hope some of this helps