I would not do Paris either. I loved going to Paris but 1 rushed day is not worth it and as others have said there is plenty to keep you busy in London. If you feel the need to get out of town I would suggest going to Windsor. I personally loved it and it was a great trip out of London.

WE loved Paris too but we spent more time in London. London has enough things to do for 10 days or more (even a month and you can't see everything).

Some Highlights that I consider "must sees" in London

Tower of London--we spent 5 hours here
Churchill War Rooms
Parliament and "Big Ben" along with Buckingham Palace--walk by
Westminster Abbey
St Paul Cathedral--worth the climb to the top
British Museum
National Gallery
V&A Museum
Museum of London
Imperial War Museum
Kensington Gardens (not the Palace interior) and Hyde Park
Hampton Court Palace--(easy day trip by train-half day venue)

I could go on and mention the little museums like John Soanne, which is unique, Courtald Gallery, etc. etc.

As you can see, you can very easily fill up 4 days and will have to chose what to leave out because you can't see all that in only four days.

Whatever you decide, enjoy your trip!

This is all a great help. Thanks everyone. We've decided to steer clear of Paris this time and focus on London, perhaps with a side trip to Windsor. You are all right, there is just so much to do in London...

Keep in mind that the Tube doesn't open on Sunday mornings until 8 am, so Sunday morning flights from Heathrow would involve other ways of getting there. We used the bus sytems at 6 am, with one transfer point, but worried that it might take too long. We've also taken the train to the airport from Paddington Station, which was easier and more expensive, but less hassle.
Also, the tube closes at midnight; make your late night plans accordingly.

Windsor is 20 minutes by taxi from Heathrow so it might be a good place to stay on your first or last night