I was born near the Cotswolds and live in East Anglia. Both are excellent bike touring destinations.
May is a great time, the vegetation is starting to green out with plenty of leaf cover. Trees are readily available in most places but East Anglia is probably better for stealth camping, more remote and rural with plenty of small woodlands along the coast and inland. I have stealth camped near Covehithe. Avoid camping close to Sizewell nuclear power plant.
The terrain is mildly rolling, with dead-flat bits from Kings Lynn to cambridge (The Fens). The local cycle touring club have literally hundreds of different routes through the dense network of quiet lanes.
I would suggest a rail journey out of London to N Essex ( a village stn), then work your way up the Suffolk coast (Woodbridge, Aldborough, Framlingham, Southwold), then head along the Sustrans route to Norwich, then up to the N Norfolk coast road at Sheringham and along to Kings Lynn, Ely (via peddars way) and Cambridge.
You cant get much better local info than that.