We are a family of 4 (48, 49, 11, 15) traveling to England for the first time. I spent a quick solo week visiting a friend in Norwich (which was AMAZING!!) recently so I've done LHR and Elizabeth line transfer to a train and feel pretty comfortable with all that. We will fly overnight arrive at 11:35 on a Wednesday in mid-June and then we need to be back at LHR the following Tuesday at 13:00 to be 2 hours ahead of the flight back.
Obviously someone could spend a month in London and not see everything so we are just going for the very top level experience - spend half a day each (different days) at the Tower and the British Museum, see Six. See the sights but not go into too many of them. I have a couple self guided walking tours bookmarked (Shoreditch and city of Westminster maybe?) and am looking at the T15 bus.
Hypothetically, I would like to take a Premium Tours private day trip to Bath, Lacock & Stonehenge and am interested in spending 4+ hours a Hampton Court.
I want this trip to show the diversity of England even though it's a such a short amount of time. Is it worth it to take time away from London proper to see these other places? I'm looking at VRBOs in London (still researching neighborhoods) and could book for all 6 nights or I am considering booking the first night at one of the hotels near Hampton Court and trying to see it on the day we arrive. From there we could go to London or make our way to Bath and catch the tour when it arrived there. We're there the week of summer solstice and would prefer to see Stonehenge without the crowds which is another concern about my timing. Maybe the Bath trip is done differently at the end of the trip to where we take a train to Bath, overnight there to see Stonehenge at sunrise and then go to LHR to be there by 13:00?