In the mid-1980s, London Regional Transport Corporation, under the ownership and control of central government (the Department of Transport), superseded the London Transport Executive, which was a department of the Greater London Council (abolished in 1986 by the Thatcher government). In 2000, the new London Mayoral regime has seen a return of public transport authority to a London-wide local government institution. What was previously London Transport (comprising the undertaking of London Underground Ltd.; and franchise arrangements for London Buses) is now Transport for London (TfL). I worked at London Underground Ltd, in the Finance and Business Planning Directorate, in 1993-94.
The London Transport Museum, or "London's Transport Museum" as it is now called under the TfL structure, occupies a statutory role, legally holding the first right of refusal for asset disposals from the London Transport Executive undertakings and their successors. This could in theory have caused problems when it came to buses, in that the franchise holders have been private companies for some years. The widened scope of TfL now encompasses hackney carriages, ie. taxis, so we hope to see black cabs (the classic Austin FX-4 model) in the Museum, or at least in the Depot (Acton) in the near future.
The National Science Museum now includes within its ambit what was formerly the York Railway Museum; now, following the amalgamation of the Clapham rail collection, the National Railway Museum, still situated in York, in expanded premises on Leeman Road, adjacent to the historic York Station. The NRM has a Friends organisation. I grew up less than a mile from York station, in fact, three hundred yards from the now-defunct York Carriageworks, which latterly built fine and reliable rolling stock such as the WAGN class 317/320s running into Moorgate and King's Cross.
The Friends of the National Railway Museum have a South of England branch, which organises meetings and social events at Marylebone station.
Going Underground is the leading "fun" site for the London Underground, and contains many links to other unofficial Underground sites that illustrate the Underground's humorous, playful, ludic quality.
See how they run has a fine selection of links to various train map/ info sites as well as a virtual Carol Vorderman page.