I've been thinking for some time about writing a few articles about unbuilt bridges, but it turns out there's so much good material elsewhere on the internet that I'm better off directing you appropriately.
The best online resource for this is the "Ian Visits" blog, which has uncovered a cornucopia of unbuilt London architecture, including bridges.
Victorian railway stations that tried to span the Thames discusses unbuilt railway bridges (and stations) at Pimlico (pictured) and Waterloo. These early proposals came to nothing, and it's only with the recent remodelling of Blackfriars Railway Bridge that a railway station has finally been extended to span both banks of the River Thames.
There have been plenty of proposals for inhabited bridges across the Thames over the years (again, some of which I've covered in the past). It's a perennial theme from incurable nostalgicists blithely unconcerned with the visual impact of what they design. One such proposal was the Crystal Span Bridge, a 1963 proposal to replace the Vauxhall Bridge with a seven storey monstrosity complete with gallery, roof garden and ice rink.
Further curiosities include a proposed elevated viaduct along Oxford Street, to allow the area below to be pedestrianised, and a high-speed travelator for London Bridge.
What unites most of these unbuilt bridges is simple: we can be glad they were never built, given the number of potential eyesores avoided!
See also Unbuilt London: Bridges To Nowhere And Mad Masterplans at the Londonist.