For more details of developments, see CULG, at
http://www.davros.org/rail/culg/northern.html ...
In summary: the C&SLR was originally King William Street (in the city) to Stockwell, extended pre WW1 to run Euston-Clapham Common (and abandoning King William Street). It had smaller diameter tunnels than the other tubes, and it used loco hauled trains.
The CCE&HR was one of the Yerkes tubes from - by WW1 - Charing Cross (now Embankment) to Golders Green and Highgate (now Archway) - i.e. two branches. It had also been (legally) amalgamated with the other two Yerkes tube railways (Baker St & Waterloo, and Great Northern, Piccadilly & Brompton) into the London Electric Railway.
So you had the C&SL running to the south via the City, and the Hampstead tube (CCEH part of the LER) north via the West End.
In 1913 the Yerkes or Underground group holding company acquired the C&SLR, but its statutory railway (C&SLR, LER, MDR and CLR) companies remained separate. In the 1920s there was a major programme of development which saw in the north, extension from Golders Green to Edgware, in the south from Clapham to Morden (my understanding is that legally this was part of the C&SLR). In between the old CS&L tunnels were rebuilt, to standard tube diameter, using standard tube stock; and to provide a perhaps more logical and balanced system, with through services from north to south there were two other extensions - from Euston (C&SLR) to Camden Town; and from Charing Cross to Keenington, with junctions between the two (C&SL & CCEH) at Kennington and Camden Town - this was presented as one of the marvels of the age, permitting non conflicting operation from both northern branches to both routes through the centre - I'm unaware of any proposal for a second southern branch until the line to Battersea (currently under construction) came about. While the LER and C&SLR remained separate companies (until formation of the LPTB in 1933), it was all operated as one. I think at the time it was considered desirable that - wherever possible - there should, and would, be through services (rather than having passengers change trains/lines). There was certainly a proposal for another interchange junction (similar to Camden) with the Bakerloo, essentially at Waterloo, in large part under the main line terminal (LSWR/SR), and the SR objected, leading to the proposal being dropped.