Time to do these tours in London?

Churchill War Rooms website says to allow 2 to 3 hours. As per RS advice, you may want to do the Tower of London Tour first on May 28 and be there before it opens.

Where do you arrive from on May 23? I am wondering if you will will be dealing with jet lag?
You have a lot planned that day and may be a little tired to do it all if dealing with jet lag.
Make sure you buy tickets ahead for important sites. There were long lines for Churchill’s WRar Rooms but we entered quickly with pre purchased tickets. Same for other London sites. We spent more than two hours each in War Rooms and Westminster Abbey.

1. The Shard on Sunday 27th would seem beyond your time available.
2. I would swap your Monday and Tuesday so you can get an earlier start at The Tower, as it only opens at 10am on a Monday.
3. Hampton Court Palace offfers considerable more to the visitor than Kensington Palace, plus Kew Gardens is close by as an ‘add on’.
4. British Library and British Museum on your arrival day (coping with jet lag ??) will be testing. Explore Soho and Covent Garden, before some time in the National Gallery, would seem more realistic.

If you have already purchased the 3 day London Pass and wish to excise decent value from your outlay ...

Sun May 27
12:00 Banqueting House
13:30 Houses of Parliament (photo stop)
14:00 Westminster Abbey (RS tour)
16:00 Churchill War Rooms (closes 18:00)

Mon May 28
10:00 Hampton Court Palace
15:00 Kew Gardens (closes 19:00)

Tue May 29
09:00 Tower of London
11:30 Tower Bridge
13:00 The Monument
14:00 St. Pauls Cathedral (RS tour)
16:00 Tate Modern (closes 18:00)

I plan to visit the British Library, British Museum and National Gallery before getting back to St. Pancras to take the Eurostar to Belgium at 17:00

All that in one day! The British Museum alone can consume those hours, plus don't forget travelling time, being at St Pancras at least 1/2 hour before your train departs, fitting in lunch and dealing with jetlag. Trying to do all three will simply mean a cursory dash through, nothing meaningful and a waste of time and effort in my opinion.

This sounds like checklist travel, and omits the time needed to commute from place to place and also the possibility of line-ups. More significantly, there's more to visiting places such as the British Museum than walking through the door. Spend half a day there and you will be able to scratch the surface. Churchill's war bunker can be understood in an hour but those unfamiliar with its background might want to spend more time. Ditto for St. Paul's. I think you should cut yourself some slack so you can linger in places that intrigue you. Slow down and smell the spring flowers.

Regarding your (IMO) over scheduled arrival day. If your train departs St Pancras at 1700, you MUST get there by 1600. The lineup just to get through security can take a very long time and then you still need to get through the passport controls. Do NOT underestimate the time it will take to get there from whichever museum you're at. And you'll likely need to add an even longer buffer to this, since I assume you'll have to retrieve the luggage you stored on arrival.

Thank you all for the information and recommendations.

As much as I love museums, there is no way my brain could absorb British Museum and the other two in one day. The British alone is a lot to bite off at one time.

For my taste, that's way too much in one day.

In my opinion, none of your plans for your arrival day are suitable. If you know from previous experience that you will sleep on the plane, I retract that comment. Otherwise, I'd suggest a walking tour (do-it-yourself or see what London Walks is offering that days) or perhaps a trip out to Kew Gardens.

Being outdoors is helpful for adjusting to the new time zone.