The Design Museum says that Harry Beck was commissioned to straighten out this mess.
Basing his map on an electrical circuit, Beck represented each line in a different colour and interchange stations as diamonds. The crowded central area was enlarged for legibility and the course of each route was simplified into the form of a vertical, horizontal or diagonal. The diagrammatic map was produced on a trial basis in 1933 as a leaflet and Beck continued to refine it until 1959. His design has inspired the maps of underground networks from New York to Sydney, and a variation of his original design is still used by London Underground today.
Other sites, including Wikipedia, suggest that he did it on his own and had to struggle to get it accepted.
Having been made redundant from the Underground Group, Beck nevertheless made a presentation visual from this initial sketch and, in 1931 presented it to their Publicity Department. It was rejected as too 'revolutionary'....Although retained by the London Passenger Transport Board, Beck remained a 'temp' draughtsman until 1937. This meant he was in the unfortunate position of being both freelancer and employee. His designer status was ambiguous and he was constantly defending his creation against badly conceived alterations.