If what you want is countryside, there's plenty of it. SwissTony suggested Kent and Sussex which are lovely counties. Consider also Wiltshire or Dorset. The Cotswolds are beautiful but perhaps a little precious and the beauty of these (and many other) counties is considerable. My feeling, as SwissTony says, is that this trip is predicated upon transport arrangements - by definition you will be going to places where there are only villages, and therefore little transport infrastructure other than local buses which will be infrequent.

As an aside, fly from wherever suits. BHX is a relatively small airport which means more peaceful, but fewer back ups when things go wrong and more limited transport links (although Birmingham Int'l station is excellent). LHR is at the opposite end of the spectrum in every respect.