There are a number of schools at Liverpool, which is a great location despite the controlled airspace and commercial traffic. Your choice really depends on whether you want to fly Cessnas ,Tomahawks, PA28s or something more exotic. My choice is Merseyflight, who as well as being a freindly bunch have a a couple of 152s and a nice diesel PA28.
There is a club at Hawarden (Chester), I believe called NWCAS, but the operating hours of the airfield are resricted by BAe, which is a bind.
Manchester International has a flying school still (I think), but the combination of the commercial traffic and the associated airspace restrictions defeat the object in my opinion.
Many people like Manchester Barton, but it is short, sometimes waterlogged, and surrounded by MAN controlled airspace.
The best "open FIR" GA airfield in an area which really lacks them is Sleap - great place to fly into, but a long drive from the cities you mention.
In truth, given the vagaries of the british weather, your best bet is probably to go to whichever is closest! Good luck and happy flying!