Transport of London, the transit service that manages the operations of mass transit around England's capital, has a convenient page on its web site for finding several transit maps in one place. Included are downloadable maps of the tube (subway), bus, light rail (DLR), and river taxi lines. There are also walking and cycling maps available, and a tourist map for finding the most popular sightseeing spots. You can find the Transport of London map page with links to all the transit maps here.
Thus far, only a map of the London underground tube system is available as a mobile app, with a list given of the specific phones for which the app is available. A link to information about the app, for which there is a nominal installation charge, is here. Elsewhere on the Transport of London site is a free signup for text messages of tube and light rail service delays, as well as highway traffic information.