score:4
The most "bucolic" area very close to central London is Hampstead Heath, which gives a reasonable impression of being out in the countryside once you're away from the perimeter and is certainly big and varied enough for a decent walk. It has reasonably villagey areas with pubs around it near Hampstead Heath station and Highgate, another pub (the Spaniard's Inn) on a road passing through it and a stately home (Kenwood House).
But you can't stay in the middle of the Heath and I think you would be unlikely to find anywhere to stay that wasn't either on a main road in an urbanised area nearby, and/or very expensive. It's always been a day-trip destination. But it's worth a try.
You might also consider Richmond (Richmond Park and the river, trips to Hampton Court). But again I think you'll find accomodation in the most scenic bits sparse or expensive.
Upvote:1
London is pretty well-connected by trains to many destinations of-note in the UK, so it's hard to take a pick. However, if you want primarily a rural experience then you have a couple of options: