There are cities throughout the UK which are longlist testers for a children's book award. Where I live is one of them. I have about 20 reading groups throughout schools in my area, where the children test a variety of books and grade them. Prior to this, I used to give my groups one book to read but I've found that taking part in this award works better, as there is an element of choice but the children are still encouraged to try different books. If you let me know the area in which you're based, I will check to see whether you are near one of the other testing centres and then can give you their contact details.
There is a group in Plymouth. If you google Plymouth Children's Book Group, you'll find out some more information. I'm seeing them on Saturday so can ask them then if you're close enough to come under their area.
My DC used to go to one like this many years ago, and it was funded by Orange - they provided stationery etc, and presumably paid for the librarian's time. Have a google and see if and big organisations are funding this sort of thing. I'd be surprised if they weren't. Try Starbucks - they collect books for children at Christmas, so might be a good match.
I have been in touch with the local library and they've been really helpful. They have said I can use the library to meet in and have even said if a organize a regular slot they will even let me use the library when it's closed so we'll have it to ourselves and it'll be nice and quiet. They've given me some advice about how to run it and said that they will help me source books if they are all reading the same thing.