The Middleton Press book "Charing Cross to Dartford" has some information and a picture of the footbridge prior to its rebuild in the 1980s. Apparently there is further information in the companion Middleton books "Waterloo to Woking" and "Waterloo to Windsor". Alan Jackson's book "London Termini" says that this link was opened in 1864 when the Charing Cross line was built.
Trains ran over this to the City, but with difficulty due to the amount of traffic on the SE Railway, and this led to the building of the Waterloo and City line with strong L&SWR support. The link line was removed in 1911 when the terminal station was undergoing major rebuilding works.