A few thoughts from a traveler who has been visiting London and other areas of the UK since the 70’s.
1.) June is peak tourist and locals travel time. Expect hordes.
2.) I disagree about Stonehenge. Everywhere in London are building cranes as older buildings disappear and modern ones take their place. Cranes are used to move materials for each story built. Cranes, tractors, trucks, fork lifts didn’t exist when Stonehenge was created. Like Machu Picchu, Newgrange, and the Pyramids it’s a marvel of engineering. If you want to see it, go. In fact take a look at www.walks.com. They do a great tour. An all day outing and since no one has yet to explain how those rocks were moved and erected I think it worth the time and effort.
3.) As noted Windsor is a good option. It’s an easy train ride from London.
4.) Hampton Court is another good option. You can use your Oyster cards for the train ride there.
5.) Portobello Road Market( Saturday is the main day) Spitafields and Brick Lane or Camden Lock Market
( next to the Regent Canal ) are all vibrant and quite the experience for first time visitors.
6.) Check the Tower Bridge lift schedule
7.) Definitely take in a theatre production. Check the Half Price ticket booth at Leicester Square for same day theatre tickets. Not knowing the ages of your children take a moment to review https://www.londontheatre.co.uk/whats-on for possible options. I throughly enjoyed Witness for the Prosecution at County Hall. It’s set in an actual courtroom. It’s been on for a while. I bought a ticket right at County Hall . Not cheap ( but that’s why one has credit cards). Secured a great seat: Stalls Row C X 55. The added plus is seeing Parliament lit up at night as County Hall is across the Thames from it. You are also next door to the London Eye and Jubilee Gardens. Lastly, you are also near the Graffiti Tunnel which is adjacent to Waterloo Station. An off the grid but enjoyable “ site to see “ especially at night.
8.) Depending on children’s ages Mud Chute Farm might be of interest.
9.) As your travel dates draw closer check Timeout London on line to see what one time events might be occurring during your timeframe.
10.) If you are going to the Tower of London arrive before it opens, get your tickets and immediately head to the Crown Jewels. After viewing them meander about the remainder of the grounds. Afterwards make your way ( Maybe 8 blocks) to Leadenhall Market. It’s the last remaining glass covered market and again depending on your children’s ages Leadenhall Market is where the Harry Potter entrance to the Leaky Cauldron is located.
11.) Visit the Museum of London as it does a stellar job of explaining how the city you are visiting came into existence.
12.) Imperial War Museum is interesting.
13.) Greenwich is a pleasant day out. Definitely take a River Cruise to get there.
14.) Check out the Mercado Mayfair for lunch
15.) Picnic in Hyde Park/Kensington Gardens
Have fun. So much to see and do. Great city!