We enjoyed a British Isles itinerary last May. Since you are on the east coast, maybe you will not be as fatigued as we were flying in from the west coast (SEA).
However, we had 3 nights to enjoy London. If you arrive one day before boarding, I would seriously consider getting a transfer from LHR to Southampton and soaking up the town, get rested and then board the next day. The fares could easily be split by sharing a car with Blackberry Cabs or West Quay Cars.
We are doing that in June for a cruise from Southampton.