You don't need to get stuck in traffic on a HOHO bus, or pay for a tour that may not be at your place.
Use the Tube, the #15 red bus, your good old feet, and even a black cab if necessary.
You can do this by yourself ...
Day 1
09:00 Piccadilly Circus
09:30 Trafalgar Square
10:00 National Gallery
12:00 #15 Red Bus to Tower Bridge
12:30 Tower of London
14:30 St. Paul's Cathedral
16:00 Tate Modern (check closing) **
19:00 Covent Garden for a typical London pub
Day 2
09:00 Hyde Park & Green Park
11:00 Buckingham Palace (check schedule for Changing of the Guard ceremony)
12:00 St. James's Park
12:30 Westminster Abbey (Holy Communion - weekdays)
14:00 Houses of Parliament & Big Ben
14:30 London Eye (or Banqueting House, or Churchill War Rooms) **
15:30 No 10 Downing Street
16:00 Horse Guards (Four O’Clock Parade)
17:00 Shopping in Knightsbridge - Harvey Nichols & Harrods (check closing times)
** Drop these out of your schedule for a less rushed day.