Sweet Thames isn't really "folk", and was written by a "Salford Scotsman", so I wouldn't myself consider it "London folk" - others have different views, though.

Beaux of London City (the Knowing Maccaroni Outwitted) is a possibility - but not much of a song! Set in St James's Park

Beaux of London City

Carthy sings Three Cripples (in London)

The first known version of Black Velvet Band was set in Barking,