Although TfL like to pretend it doesn't exist, Thameslink has been running through central London for 30 years now, (and like the RER the original rollinmg stock has been replaced) and it has several hugely-expanded stations (see Blackfriars and London Bridge in particular, but not forgetting the new stations at St Pancras Low Level and City Thameslink, which replaced Kings Cross and Holborn Viaduct).
Yes, it's only one line whereas the RER has five. But many of our Tube lines also go much further out than the Paris Metro does, (e.g Amersham), and also have taken over main line suburban routes to West Ruislip, Harrow & Wealdstone, High Barnet, Epping, Hainault, Upminster, Wimbledon and Richmond, (and, arguably, Stanmore) in some cases more than 100 years ago.