Three Londons: The City, Greater Westminster and Southwark

Have, at least until 1965 (bonus points if it lasts beyond,) there be three separate cities in what is OTL known as Greater London, those being The City of London, Greater Westminster (in French:Ouestmoutiers) and Greater Southwark across the river. Westminster thus becomes the name used by the public to generically refer to what's north of the river while Southwark is likewise for the South.

Have it be not just a question of names but one of municipal government. Have the cities share overground and national rail lines but have Southwark develop its own underground railways but call them the Subway and have trains travel on the right to be different from those of Greater Westminster. On the whole, have Southwark be as different and as independent as possible from Greater Westminster.
 
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