I’ve been leafing through the official guide and it does include towns/villages with campsites on the facilities table (pages 92 & 93). There’s not masses but you could plan a 2 or 3 dayer around stopping at ‘legal’ campsites. The guide makes no mention of wild camping that I could see so CUK are giving a “responsible steer” towards more conventional accomodation.
On top of that, Bramshill (mentioned in the FB comment above) is pretty close to Wellington country park, which has a woodland campsite (apparently re-opened mid-April).
I think given the fact you’ll be trundling about semi-rural Berkshire and Hampshire, discrete wild camping is inevitably going to be harder. There’s lots of grumpy old duffers down this way. But also don’t shit in the woods (and leave it there), you’re not a bear…