If you asked 10 people you would get 10 different answers, although I would be impressed with someone who can name 20 schools for commercial flying training in the UK.
You will be faced with a question:
Integrated: do all the training at the one school on a zero to ATPL licence
Modular: Pick and choose elements of the required course at separate un-connected schools.
Hence, for Integrated there are only 2 in the UK: CCAT and Oxford.
For modular there are quite a few. Some just do the ground school, others do just the flying bit, and others do both the groundschool and the flying bit, but are still modular. Both the above integrated schools also do modular courses.
I'm doing the modular method, and I selected Bristol Ground School for my ground studies. I could have chosen a host of others, but I mainly went from recommendations from people I know and from this site. I'm quite happy with my choice. I have not yet selected any for the flying, but my shortlist is PAT, Bristol, Atlantic, Aeros and Westflight.
Do a seach on this site and it should become clear which schools are recommended. However, I would urge you to visit a school in person. I even went to visit BGS!
You could also look at schools outside the UK. There are a few in Europe, and quite a few in the US.
As for your private pilots licence, you can do that just about anywhere, and most would select their local flying club.