There isn't just one that mimics (near as dammit) the hop on hop off services, but rather a selection of several.
One way to start the journey would be to take the Underground from Earl's Court to Westminster. After having a look around there take the number 11 towards Liverpool Street station to Charing Cross for a 15 (or 15 Routemaster) towards the Tower.
The 15 passes the Royal Courts of Justice, St Paul's and the Monument. From Tower Gateway take the RV1 towards Covent Garden along the South Bank, past the Eye and to the Aldwych.
From there you can take one of the buses along Kingsway to the British Museum or the 9 to Green Park for a walk along Constitution Hill to Buckingham Palace. From Hyde Park Corner take the 74 towards Putney back to stop A in Earl's Court Road.
tfl.gov.uk/tfl/…westminster-2298.pdf
tfl.gov.uk/tfl/…charingcrossquad-2048.pdf
tfl.gov.uk/tfl/…aldwychquad-11500.pdf
tfl.gov.uk/tfl/…toweroflondon-2261.pdf
tfl.gov.uk/tfl/…countyhall-2060.pdf
tfl.gov.uk/tfl/…hydeparkcorner-2140.pdf
tfl.gov.uk/tfl/…earlscourt-2070.pdf