Just as a reminder that Hampton Court is actually very close to LHR. You can even get there by bus directly from the airport and vice versa. However, to get there from central London takes considerably more time.

The British Museum can be combined with other attractions in Bloomsbury and surroundings like the British Library, various University of London buildings and exhibition galleries and of course the famed bookshops at Charing Cross Road. That would make a full day, the studenty pubs in the area are quite fun in the evening time.