I was going to suggest the Camping and Caravan clubs.  Although no experience of them I know they have an excellent reputation for their sites.

Your idea does remind me of a holiday in my teens with my parents, where we toured in Dad's old Hillman car.   We started by following the Thames to it's source, and then we went to Herefordshire and nipped across to the Gower Peninsula to meet up with family, and then gradually worked our way through Somerset (Wells, cheddar etc) down to Devon and Cornwall and then back along the south coast as far as Hayling Island, before making our way home to North London where we lived.  We didn't book anywhere at the time and we stayed in bed and breakfasts everywhere.   Although many years ago now - I can still remember that holiday, and it was a tad strange having to find a different place to stay each night.   I know we didn't book in advance at the places we stayed although Dad did plan an itinerary of our route for the two weeks.

I bet you will have loads of fun, and definitely a good way to see the country, and I expect you will have great fun planning where you want to go and what you want to see along the way.

Mags