Just to be clear, it could be a possibility but still crazy. Kind of a tough call...
Your enemy is opening times of attractions. And travel. Where is your hotel?
This is subject to change, but the last time for entry into the Tower is currently 1530 and the grounds close at 1730. Doing the HOHO and getting back could be tough. It might be the one time I'd suggest getting off the bus, touring the Tower, and finishing the loop. But you're likely to be exhausted, even if flying in the front of bus with arrival lounge access...
Westminster Abbey is open from 930-1530
Churchill War Rooms are 930-1800
Changing of the Guard begins around 1030 and lasts about 45 minutes.
It's about a 10 minute walk from Westminster Abbey to the Churchill War Rooms. Google Maps says 15 minutes from the Abbey to Buckingham Palace. It always seemed a lot longer...
Your only realistic ability to visit them all would be to start at Buckingham Palace. A brisk walk, allowing for the departing crowd, to the Abbey for an 1145 ish? entry. The guided Verger tours can only be booked on site. Those take 90 minutes. Problematic if you're trying to see the Changing of the Guard on the same day.
The Churchill War Rooms will take as much or as little time as your personal interest in Churchill and WWII. It's a fascinating exhibit/museum. We spent at least a couple of hours there on our last visit, but I'm retired Army with a lot of time working around NATO and a WWII history buff. But they're at least an hour.
As long as you're dealing with someone who does the London - Windsor - Southampton thing regularly, that's probably doable. I kind of like the idea of doing it after the cruise, and ideally stretching one last day so you can see some of Windsor and maybe wander over to Eton, but that type of tour is done enough that you should be fine.
Can't help with preferred HOHO company. It's been a long time and I know there are at least a couple of them.