I'd go for Windsor; not very far from the airport--you can get there by train (Heathrow Connect to Hayes & Harlington; then connect to any westbound local train to Slough; then hop on the shuttle train to Windsor & Eton Central).
I thought Windsor was one of the best places I've been in the UK (and I've been to a lot of good places there). The castle is fantastic; and the town has plenty of shopping and restaurants.