Ah, at last someone names places.
New Forest - train to Southampton, Brockenhurst, or Lymington. Then use buses such as #6 Lymington to Southampton or the X7 to Paultons Park. For handling areas with fewer buses, taxi from Lymington to Beaulieu is £14. It's a holiday, so take the ferry to Hythe. We've holidayed happily car free.
Devon - Train to Exeter, Plymouth, Barnstaple etc. Buses like 21A and 310 would get you around sights near Barnstaple. Exeter train to Dawlish etc. We've holidayed happily car free.
Cornwall - haven't been for years but have been years ago car free stopping along the train line, bus to Eden project etc.
Cotswolds has trains throughout; needs a bit of planning to work out the best bus or train service sometimes, but everywhere was covered easily by looking up routes each evening for the next day, and two short taxi rides when we wanted to connect more quickly. The steam train from Gloucester was a nice way to get around while enjoying the journey.
Dorset you again pick a side based on train connections and travel round a hub; Bournemouth or Yeovil or Weymouth. We haven't been post child nor post covid, but it was fine for us both car free before.
Now, those who are determined to live 8 miles from a village, 24 miles from the nearest school and work 3 towns in the opposite direction - yes, they will certainly struggle if they don't drive in some of these counties. Those are all clear choices some individuals deliberately made though, in full knowledge of their options for getting around, and certainly it isn't a large proportion of people who make their own lives difficult.