Hallo there,
I know it's not an answer to your question as such, but I have to plug the RA of my home county, Northumberland. Not just 'cos it's good examples, but also because one of the main dense patches is totally accessible by a combination of bus and walking. I don't have a car either, but have found that the buses from Alnwick to Wooler go right past a number of good sites, Old Bewick, Chatton & Ketley, Weetwood & Fowberry and Coldmartin & Doddington.
If you could be arsed to do the train to Newcastle, you could be up Chatton in an hour and a half on the bus. And there are a couple of decent low cost campsites in Wooler, and even the Bunkhouse at Chatton if you weren't keen on a tent.
This message was totally brought to you by the Northumberland Amateur Tourist Blethering Association