Dan: Are you planning to go inside Buckingham Palace to take the tour, or are you just interested in seeing the outside of Buckingham Palace. I ask because I bought the timed tour ticket to go to the Queen's Mews, the Queen's Gallery ( paintings on display) and then inside the palace to see all the rooms that are open to the public. This was a whole day event for me. You could go inside the palace but not spend a very long time ( I did spend a while inside, the docents were excellent and it is a self guided tour). The palace is massive and the rooms are huge and beautiful. I loved the tour and it was really worthwhile. Seeing the inside of Buckingham Palace is very impressive. This of course depends if the palace is open if for vistors due to the Covid-19.
If you are doing that, you will not have time to see the changing of the guard ( due to needing to go at your timed tour and should start out early for it). You will also not have time to visit Kensington Palace and see the inside of it. You could finish the day with going to Harrod's or Trafalgar Square or one of the nice parks if the weather is good.
Victoria and Albert is massive, allow time to see several galleries.
When I went inside St. Paul's I bought a timed ticket, do allow at least an hour to see the inside of the church. You may also want to go to the top of the dome, I did not.
Do go to the Tower of London first thing in the morning and head straight to the crown jewels. I went into every single tower and saw it all. I have been there about four times and the last time I saw it all. I have walked across the Tower Bridge so allow time for that.
Are you planning to go inside Kensington Palace, that is lovely. I went inside and again I had a timed ticket. The gardens are lovely and there is a large pond and walking around the outside of the palace is nice.
Others are right, do get a timed tkt for the Churchill War rooms and allow at least two hours to see it. It is full of information and it is really interesting. I have been there twice, years apart and it was good to see again. It is a very popular museum.
If your at the National Gallery and if there is time, St Martins in the Field church is right across the street to pop in and see this lovely church. Small, not a massive church. It also has the crypt where you can get lunch or dinner, cafeteria style but very nice.
If your play is at the Globe theatre at 6:30PM, leave time to get there from the National Gallery. It is on the other side of London. If you were attending a play in the theatre district, then it would be close to the National Gallery.
My point I am trying to make, all the things you want to see are amazing and take time to see. So you may not get to everything you want to see on your list for that day. I would recommend going on each website to buy timed tkts.
Oh, by the way around the corner of the National Gallery is the National Portrait Gallery which is also interesting.
Hope this helps.