I have to ask you first the typical travel agent's questions, Hannelore.
What time of the year and how long for? Personally, summer and spring are the best times to come to London when festivals (most of which are free) in town adds a tremendous lot to enjoying the city and the people.
As to permanent tourist fixtures, you'd be better off having a personal guide...ME!!!..than going with a group sightseeing tour. I would, however, take you to explore the city on foot. Depending on duration of stay, I can show you as much of the city as you can walk it in a day.
Day 1:
Within an 8-mile radius and do-able in a day: Starting point will be Trafalgar Square towards Whitehall, on both sides of which will be Banqueting House, the Cenotaph, the War Rooms, Downing Street. At the end of Whitehall is Parliament Square where you find Big Ben, the Houses of Parliament (which you can visit and watch democracy in action. Tickets are applied for with our local MP's, so, you'd have to know someone to have access to that)
Then, proceed to Westminster Abbey, cross the road to Buckingham Palace * (which is only open to the public in the summer time). The changing of the guards takes place at 11:00 A.M. so, if you want to see them prancing about in bearskin or redcoats, this walking tour can be done counterclockwise. A meal of fish and chips in-between, a very typical British fare to be washed down with lager or cider, if you like warm beer, that's on tap too.
* Trooping of the Colour is in June. You may catch a glimpse of Elizabeth R and the Royal gang as they process from the Palace down the Mall to Horseguards.
Day 2:
East end of London starting from Covent Garden (although Eliza Doolittle no longer hawk flowers there) St Paul's Cathedral (you can walk the aisle where Diana processed for her wedding or sit by the steps and sing "feed the birds tuppence a bag"), Tower Bridge (the fancy one tourists often mistake for London Bridge...the original which is in Arizona BTW), Tower of London (the Crown Jewels are there, and site of many beheadings under Royal Warrant...Anne Boleyn for one). Tea and scones at respite. How very British, eh?
Go for a Shakespeare play in the summer at The Globe or
Jack the Ripper Walk at night (if you go for that sort).
Day 3:
Cross the Thames River to the London Borough of Greenwich (where I live and 7 miles out of the city centre). The Greenwich Meridian Line, Royal Observatory, Cutty Sark (it was partly burnt down a few months ago but, should be back to form soon) the Millenium Dome (now O2 Arena), Queen's Palace, which was the Monarch's residence prior to Buckingham), the Naval College...respite at Trafalgar Tavern overlooking Thames River.
Night-time: A West End show. British production, of course.
There is so much to see and do in London. I could go on and on but, rest your feet for now. Geesh, I should send a fee note to the London Tourist Board!