As already suggested reserve tickets to tour the Palace and Churchill War Rooms.
Then group your MUST sees: Opening and Closing Times will play a roll in your ability to see things.
WED:
9:00am-Breakfast at St John Bread and Wine at Old Spitafields Market.
Walk from there to see the last glass covered market in London, Leaden Hall Market. From there walk to see St Dunstan in The East Church Garden.
Then off to the Museum Of London which opened at 10:00am. 25 minute walk.
After seeing the Museum ( excellent choice by the way) time for a late lunch. Walk over to One New Change. Plenty of options for lunch there and you can take the elevator to the roof for a POV of St Paul’s.
By now it should be close to 5pm so attend the 45 minute Evensong at St Paul’s.
That evening explore your neighborhood and enjoy a relaxing dinner.
THURS:
Breakfast at Beas of Bloomsbury in Russell Square
Then off to see the British Museum which opens at 9:30am.
From there walk to the British Library to see the Treasure Room. Then it’s time for lunch.
Try Dishoom at Granary Square which is behind St Pancras. Or checkout the adjacent Coal Drops Yard for a lunch spot OR if someone wants tea or a cocktail in a gorgeous setting visit the Booking Office at the St Pancras Renaissance Hotel. OR you could walk along the Regents Canal to Camden Lock Market to have lunch and shop.
If you do the latter then hop on the tube at the Camden Town Underground Station and take the Northern Line to Waterloo Station. 15 minute ride.
Exit Waterloo Station and walk over to Jubilee Gardens and ride the Eye.
FRIDAY:
Start the day with the Buckingham Palace Tour then walk through St James Park to the Churchill War Rooms. If you are looking to return to Westminster Abbey this would be the day to do it.
After the War Rooms or the Abbey walk up Whitehall to Trafalgar Square and the National Gallery.
From there walk behind the Gallery to Leicester Square to the 1/2 ticket booth and get tickets for theatre. Then one goes to the Tate Modern, the other doesn’t. Maybe Imperial War Museum for that person.
Plan to meet back at hotel in Kensington and off to dinner before your choice of theatre.
SAT:
Go early 8:30am to Portobello Road Market.
Tube to the Ladbroke Grove Station walk to the market, meander, shop, have coffee and a nosh. You’ll miss the hordes if you go early. After waking the entire length of the market walk out to Bayswater Road and turn left. Walk about 6 blocks heading towards the Queensway Underground Station. Just before you get to it cross at the light across Bayswater Road to Broadwalk and walk into Kensington Gardens. Princess Diana Memorial Playground will be on your right. Head to the Round Pound and Kensington Palace. Photo ops galore.
From there walk to the Albert Memorial and then over to the V & A. You can easily spend hours there.
At this point a pub lunch should be considered. There’s a small pub in the mews that I like for lunch or pint. The Queens Arms.
From Queens Arms to Harrods.
That night get a good night’s rest.