If you are in Chelsea just take the regular 11 bus. No commentary of course, but will probably go by as many sights as many of the tour buses circular routes.
Starts close to Fulham Broadway station in Walham Green. Swings immediately to the Kings Road as it makes its way through Chelsea, passing close to the National Army Museum and the home of the Chelsea Pensioners to Sloane Square where another dog leg takes it round to Buckingham Palace Road, and Victoria Coach and railway station.
Up Victoria Street passed the mixture of shops and offices, many of which are government buildings including New Scotland Yard, and Westminster Cathedral. At the top make a left hand turn at Westminster Abbey round Broad Sanctuary onto Parliament Street and Whitehall, passing more 'historic' government buildings including the Foreign Office and Ministry of Defence, the entry to Downing Street, Horseguards Parade etc. (The Churchill War Museum being behind the Cabinet Office).
A 270-degree right turn at Trafalgar Square passed the National Gallery and up the Strand passed Charing Cross Station, round The Aldwych, passed the Royal Courts of Justice and into Fleet Street where the ghosts of some of the old newspaper buildings such as the Daily Express are passed, plus all the pubs the journalists used to go in all afternoon and the road to The Old Bailey before reaching St Paul's Cathedral up Ludgate Hill.
Continuing through the City passed the Mansion House, the remains of the Temple of Mithras (go back to Museum of London near St Paul's to see some of the artifacts) onto the Bank of England, through the financial heart finishing up at Liverpool Street Station (or nearby with all the Crossrail construction).
For best views overall, sit on the left.