I'd suggest York rather than Cornwall, especially in winter. You can assume some cold wet weather and it's easier to manage that in a city like York than a spread-out mostly rural area like Cornwall. The best part of Cornwall is the scenery, best enjoyed in warmer dryer seasons (dryness is relative, I know).
York is also much easier to reach from London, a two-hour train from Kings Cross. The city has a great cathedral, museums, the old wall, a compact pedestrian center, lots of good eating and shopping, pubs etc. You could easily spend two full days there, or three if you're enough of an old train nut to spend a day at the Railway Museum. No need for a car unless you want to drive elsewhere, and there are good options for day trips too.
Another option, closer to London, might be Oxford -- a good-size city with lots of compact sightseeing, shopping, dining etc. options, about an hour from London by train. Nearby Blenheim Palace could be an easy excursion.