Heavily depends on where and how rural. For example I lived in Lincolnshire where there were villages that had everything you needed, good walks and sports/social facilities. Same for Buckinghamshire. Wiltshire on the other had, muddy, rainy and dank for 11 months of the year. Oxfordshire also good.
General downsides:
Mud in winter. Offset by good wellies you'll live in. Constantly having to clean car and add screen wash because of mud. You need a mud room if you have pets or kids.
Agricultural impacts: smell, flies, noise, waste, harvest time causing a huge influx of thrips that get everywhere including up your nose and in your eyes. Gluts of particular insects every year. One year hoards of hover flies, another year, hoards of ladybirds etc.
Difficult to actually get the lovely food you see growing in the fields as it goes to posh London restaurants.
Side roads not de-iced or snow cleared leaving you stuck.
Poor internet.
In some areas (not Bucks) lack of footpaths or maintenance of footpaths and a surfeit of farmers with guns to tell you to get off their land if you set a toe off the badly maintained footpath.
Depending on where you are rurally loud aircraft noise and low flying from the RAF.
Crap bus services
In some areas lack of mains sewerage and gas
Upsides:
Fresh air/less pollution
It's really lovely seeing the seasons and you get to know all the local flora and fauna.
Aircraft and low flying if you like that!
Good for bike riding
Non chain pubs that serve great food
Cheaper stuff in some places
Some great mobile NHS services