Not sure how long you will be there, but I'd spend 4 days in London and 3 days to the West. The must-sees in London include:

• Westminster Abbey - and walk around the Victoria Embankment and Westminster Bridge, Parliament, Big Ben, London Eye, etc.
• Tower of London
• St. Paul's Cathedral and the City of London
• National Gallery and Trafalgar Square
• British Museum

With 4 days in London, you can also likely fit in the changing of the guard and Buckingham Palace, Churchill Museum and War Rooms, Hyde Park and Kensington Palace, Covent Garden, theatre, British Library, King's Cross station, etc.  

Then, rent a car from Heathrow and drive west. 1 night in Bath, 1 night in the Cotswolds or Stratford-upon-Avon. Along your way, consider to visit:

• Windsor Castle (to or from Bath or day trip from London)
• Stonhenge and Avebury (on the way to or from Bath)
• Highclere Castle (setting for Downton Abbey - if you were in to that)
• Salisbury Cathedral (hour SE from Bath)
• Wells Cathedral and Glastonbury (hour SW from Bath)
• *Roman and medieval bath houses in Bath
• Costume Museum in Bath (better than you think)
• Museum of Bath at Work
• *The Cotswolds (between Bath and Stratford) - Chipping Campden, Stanway House, Stow-on-the-Wold, etc.
• Warwick Castle (20 minutes north of Stratford)
• Shakespeare in Stratford (not sure whether plays run during Christmas time!)
• Bampton (setting for Downton Abbey
• *Blenheim Palace (about an hour south of Stratford, 10 miles north of Oxford) 

* Don't miss!