Many of the tourist sights such as Piccadilly Circus, Trafalgar Square (which are mainly intersections of traffic, except Trafalgar has the black lion statues and Nelson's Column), Houses of Parliament & Big Ben, and Buckingham Palace CAN be just drive-by sights if you are pressed for time, which you are. You are in the position of wanting to see a lot in a short period of time, plus include a trip to Hampton Court. To actually go into each and every one of the sights, such as the Houses of Parliament, would take a tremendous amount of time. Hyde Park, St. James's Park are great for a walk, but it's up to you if you want to spend time there. That's why, for some places you have heard of, a drive by and a good look may suffice for this trip.
In my opinion, a Hop-On-Hop-Off bus tour could be a very good idea for you. The buses do a 2-hour loop around the tourist sights, with narration as to what you are seeing, and the history of each sight. It is a fantastic way to get an overview of all of London in two hours. It will help you to see it all, get a glance at everything, pick the things you want an in-depth visit to, then spend the next two days visiting the sights. Your ticket is good for 24 hours, and during that time, you can ride the bus as much as you want. You can ride, hop off, go see a sight, hop back on and ride the loop awhile. This ticket also includes a boat ride on the River Thames. This is a great way to see the landmarks in London from the river.
John, with this new information: "Hampton Court is an absolute must", here is a new schedule for you:
Nov. 3--Only 1/2 day, or 1/3 day.
Arrive in London, get to hotel from airport, get checked into hotel, freshen up in room(s).
Get out for a walk or sightseeing, to avoid jet lag.
Buy a hop-on-hop-off bus ticket for each person. Do the two hour loop around London.
Do one quick sight, such as Harrod's.(Because the ladies want to see it.)
Take the river boat ride that comes with your HOHO bus ticket.
Use your hop-on-hop-off bus ticket to get across town to the Tower of London.
Get to the Tower of London first thing in the morning, to avoid crowds.
The Tower has a good lunch cafe when you are through with sightseeing at The Tower.
Take bus to St. Paul's.--If you have chosen not to buy a HOHO bus ticket,
then you will take the Number 15 bus from Tower to St. Paul's.
Then bus or tube to Westminster Abbey.
Leave for Hampton Court early in the morning. Take train out. It will take 1/2 day or 3/4 of a day to see it properly.
Hampton Court has a cafe for lunch.
Take train back to London.
Museums you may want to choose from that afternoon include:
The British Museum
The National Gallery
The National Portrait Gallery
Museum of London
***Pick and choose from this list which you want to see. Meaning, a couple of them, not all. If you don't like museums, don't go to any of them.
This would be the time to visit other attractions that are "must sees" for you, such as Churchill War Rooms.
Maybe some combination of David's list and my list will be helpful to you. Have a great trip!