What order to do these things? Suggestions for anything else?

Relegated, it would help if you would use your actual dates versus days of the week, and which attractions you plan to just walk by versus visit inside? It makes a difference which of them you should purchase advance tickets for.

Added dates and if we would see/tour each one.

I bet your kids would love to climb the Monument. Great fun. Do you know the importance of its height relative to its position? Pretty decent views too. You might consider it.

Yeah I saw it on some YouTube video that it was the distance form the fire, I think it wouldn't take too long on our way to the next thing on the list.

Tower Bridge: We enjoyed walking across the glass floor. Great views, interesting engineering.

Below is an updated plan, tried to group things together by their general location. Not sure if this is trying to do too much per day.

Tuesday July 30 LONDON (DAY 1)
9:00 AM Breakfast
10:00 PM Buckingham Palace
11:00 AM Westminster Abbey (tour)
12:30 PM Lunch
1:00 PM House of Parliament / Elizabeth Tower (Big Ben)
2:00 PM Churchill War Rooms (tour)
4:00 PM Thames Rocket
5:30 PM Dinner
7:000 PM London Eye

Wednesday July 31 LONDON (DAY 2)
8:00 AM Breakfast
9:00 AM Borough Market
9:30 PM Tower Bridge
10:30 AM Tower of London (tour)
12:30 PM Lunch
1:30 PM Monument to the Great Fire of London (tour)
3:00 PM Sky Garden
4:00 PM Globe Theatre (might not do this)

Thursday August 1 LONDON (DAY 3)
8:00 AM Breakfast
9:30 AM Go Ape (Battersea)
12:00 PM Lunch
1:00 PM Natural History Museum (tour)
5:00 AM British Museum (tour)
6:00 PM British Library (brief tour)

I think 2 museums in 1 day is too much, even for people, like me, who love them. I think St. Paul's definitely worth doing, especially the Whispering Gallery.

On Wednesday you are going to Borough Market why not have your breakfast there? Secondly you are so close to the Shard why not taking in the views from there before you walk across Tower Bridge to the Tower of London. Is your Tower tour at a specific time? If you went to The Shard then you can forget the Sky Garden.

And I agree don’t miss St Paul ‘s.

I’d move London Eye to Tuesday as your “ in the neighborhood.” Makes no sense on Thursday as neither the British or Sherlock Holmes museums are nearby.

Also agree two museums on one day would make my eyes roll back into my head but it’s your schedule. Might I suggest breaking it up with a nice lunch and a stroll in Regents Park.

Enjoy you time in London. Fun city to experience!