We were just there at the beginning of June.
Churchill War Rooms - definitely pay for the guided tour, if it's available.
We LOVED our early entry tour of Tower of London with Walks, here:
https://www.walksdevour.com/london/tours/tower-of-london-guided-tour/The closing ceremony tour looks really cool, too.
We also did Westminster Abbey and Changing of the Guard with them, but it wasn't as interesting/engaging as the Tower of London. We did enjoy Changing of the Guard, but be aware the ceremony start to finish takes like 1.5-2 hours. We just watched the first part with the parade.
Also enjoyed the river cruise we got as part of one of the tours, but I would pay separately to do it, too. We rode all the way from Westminster Abbey to Greenwich (we were meeting someone there).
We also really enjoyed the self/guided tour of Tower Bridge. Walking across with the glass floor is fun.
Borough Market is a good place to get lunch, if you want something quick and your group has diverging tastes. Nothing we had really blew me away, though. It was fun to walk around (although very crowded).
We opted not to do the Harry Potter studio tour as a coworker said it's basically an all-day event. Quite a lot of travel time on a bus to get there and back.