To your first question, it is possible to get to both the Lake District and the Cotswolds by train, but if you don't have a car you'll be limited to buses (or walking) to get out into the country in these regions to get that "village-y" feeling.
Personally I felt that the Lake District was somewhat overrated unless you're a serious walker or a big fan of the 19th-century literary figures associated with it. The town of Keswick seemed a little overly cute, and the TI office was very busy even on a weekday in mid-May. Also, I guess my impression was influenced by the fact that the public restrooms aren't at the TI office, but rather a few blocks away at the Pay & Display car park, and not very well signposted -- I had to ask a couple of people in order to find them, while I was becoming increasingly desperate, LOL.
Can't speak for crowding in the Cotswolds, but another region that's equally accessible from London, or even more so, is the East Anglia - Norfolk region. That might be more relaxing. Check threads on this forum about places there if you're interested in that as an alternative.
There's also Cornwall, of course, but it's also likely to be very busy in June and it has similar limitations with public transportation.
Yours is a typical dilemma because train and bus lines tend to go where there are plenty of people (i.e., paying customers), thus the smaller village areas don't tend to get convenient or frequent service -- or any at all, depending how remote. For example, train service to Marlborough in Wiltshire (near Avebury) was discontinued decades ago. I would say an exception is in Scotland, where the buses go to places that feel amazingly remote.