I'm afraid I completely disagree with Vfe.
Size/ reputation matters. If someone said you had 4As at 'A' Level and you could go to Oxbridge or John O Groats technical college to get a degree it is ostensibly the same question. When you finish you can choose where to work from the big schools to the small clubs but the reverse would probably not be true. I am not saying this is how it should be I am just saying this is how it has panned out in my experience.
You should always get the best facilities, most up to date methods/doctrine and best aircraft possible for anything in aviation be it PPL, or MEIR then you can work at any level less as you never know where you are going to need work in the future.
A small local club is unlikely to turn you down because you went to Oxford. However the big schools will often put you behind others if they feel they trained at somewhere with more pedigree.
There are some centres which are not big schools but would be considered centres of excellence for instructional courses and search for Lady in Red will help you find examples of such. Suggest you want a busy/difficult airfield for FIC if at all possible i.e. Cardiff, Bristol, Bournemouth, Exeter, EMA, Leeds, EDI, etc as this will give you better prep for dealing with all types of instruction then if you go to small or grass airfield you will have the complete set of skills wrt airfields.
There are some definite places to stay away from but you don't live near Luton
Good luck