The following is mostly lighter reading not classics as such.
I enjoy Miss Read for light reading focused on characters in small English cities or villages.
Edith Pargeter has some good character heavy books written under that name (I particularly enjoyed
Lost Children which is set in post-WWWII England).
The semi-autobiographical Candleford Trilogy by Flora Thompson(omnibus titles
Lark Rise to Candleford provides a fascinating look at being rural poor in late 19th Century England .
I've heard good things about
The Provincial Lady by E. M. Delafield and the Barsetshire series by Angela Thirkell.
The Blog
Furrowed Middlebrow has reviews on many books that fit your description some of which are actually being republished.