Is Pickering too far out for you? We went to Vale of Pickering for the May bank holiday and really enjoyed it. Loos are good and v. clean, excellent kids play area, big field for ball games, etc. You do need to use the car to get about though, as it was quite a trek to the nearest village.
We struggled a couple of years back to find a site in Derbyshire... can't remember the name of the place we ended up but I wouldn't really recommend it anyway! It was very basic and very exposed to the elements - not good in gale force winds and torrential rain!!