I have never heard of a "free city bus". Depending on your flight times (ie, not too early, not too late), you can be in South Kensington in 35 mins on the tube from LHR (for about £4). From some of these stories I have heard about transport to/from the LHR hotels, I am sure I've been in my hotel at Gloucester Rd tube before people staying at the airport were in their rooms.