As the title states, why was third rail mostly restricted to South Eastern, Southern and South Western railways? Was it because other regions adopted diesel to replace steam? Was it down to the timing of the replacement of steam, i.e. did the early converters use diesel and later ones use electric, or vice versa?
Early conversions from Steam used Direct Current electric both overhead (mainly 1.5kV) and DC third rail (mainly 750V). It was more suited to intensive suburban services so did not go far out of London or other cities (see below).
The overhead you see nowadays is Alternating Current 25kV. The technology for this did not exist as early as DC did and the UK was one of the last companies to adopt AC as standard in the 1960s.
The southern DC did not extend far out of London. The equivalent to the North of London was underground (not much goes very far South). So there was very roughly an even spread of DC third (& fourth rail) electrification in both North and South London suburbs.
Almost 100% of the Southern region (SE, S and SW) eventually got third rail. This was as Steam got replaced and the third rail got extended further and further until it reached the channel coast. It would have been AC but the London end of the lines was DC.
Some aberrations :-
1) South central had some DC overhead but that got changed to third rail long before nationalsation (1948).
2) Some southern freight yards had overhead (presumably 750V DC and used by class 71). This was to make life safer for staff. Electro-Diesels (class 73 & 74) obviated that need around the 1960s-70s.
3) Lines out of Liverpool Street were iirc originally 1.5kV DC but were converted to AC around the 1960s.
There was DC third rail outside London.
1) Liverpool had suburban lines that are DC third rail and this still survives. They Currently use class 507s and identical class 508s from Southwestern. These are about to be replaced.
2) Manchester had some with side contact. This is now a tram route.
3) Newcastle had suburban lines using DC third rail. These reverted to diesel in the late 60s/early 70s iirc. The units came to Southern.
4) Manchester to Sheffield and Wath had DC in the form of 1.5kV overhead. Most of the route got closed and the Manchester end was converted to 25kV AC.