Take heart, it should work. Wife is Japanese. Moved back from Tokyo to live in very rural area few years ago, for same reasons as you. Has gone very well. Wife found other Japanese females living locally very quickly (maybe Facebook, and forums for Japanese overseas?) and started meeting up with them within week or two. Probably has more friends that me! Kids have been absolutely fine, acclimatised very quickly, were roughly same age as yours.
Feel that it's less an issue of Japan vs UK and more urban vs rural. If your wife likes countryside, gardening, outdoors, and realises compromises that village / country life entails, fine. If she vaguely expects there to be a conbini within 200 yards of the house, thinks she'll get decent shopping and swathe of nice restaurants locally, etc. then might struggle. (Though Cotswolds probably pretty good for food I would think?) If she is committed urbanite and not that keen on countryside at all, then probably going to be tough, at least initially.
Tip: whatever outside weather, always having a warm house helps - "just put a jumper on" isn't their culture, as you may have noticed. Use CH lavishly, insulate where you can. Living here I find weather often great (April and May this year!), but unpredictable / inconsistent.
Not actually cold - little frost, almost no snow, so better than almost any place north of Tokyo - but must be ready for anything, every day. Be flexible. Precipitation in Tokyo much higher than that of London, but concentrated at different times of year. Summer anywhere in UK far less enervating than that of Tokyo. Good summer day in England countryside better than almost anywhere - not so hot that you can't actually do anything, little humidity, cool in shade.
Sum up: different life, believe better for mental health of adults, better for children in my view. Largely depends on attitude - positive or negative, you choose. It is a choice.
PM me if you like.