3rd rail south but overhead north

A one word answer would be "history"

To elaborate, most of the decisions were taken before nationalisation (1948) as the Southern companies had decided to "do their own thing" many years earlier. (There is a lot on wiki re Southern Railway companies' electrification). Other companies were much later players with electrification, and I think 3rd rail was just simply not in favour when later decisions were made (exception: L M S Euston & Broad Street services)

There is MUCH more meat for others to put on these bones for you as their knowledge is superior to mine.

High voltage systems are better suited to long-distance transmission but have to use overhead cables for safety reasons. Low voltage systems are cheaper to install (no modifications to bridges and tunnels), although substations need to be closer together because of the greater transmission losses. High voltage systems also need more on-board equipment (transformers etc, to get down to traction-compatible voltages) so are proportionally more expensive when used on intensive services.

The choice of system by each company depended on the preferences of the companies' electrical engineers and their aspirations. So the LSWR wanted to biuld up a suburban network as quickly as possible, and went for dc 3rd rail. The Met originally wanted to use overhead, whilst the District wanted a 4-rail system: a parliamentary commission had to arbitrate and found in favour of the District. The LNWR's London area operation was integrated with the Bakerloo line (part of its raison d'etre being to take local traffic out of Euston), so had to use the same 4-rail system. The NER, electrified some of its network on overhead. The LBSCR, with aspirations to electrify its main lines as well as the suburbs, started on an ac overhead network, but after merger with the LSWR to form the Southern the (largely ex-LSWR) management standardised on the LSWR's dc system (which, being both simpler to install and not reliant on German suppliers, had become much more extensive in the decade preceding 1923).
The LNER started a number of 1500V dc overhead schemes in the late thirties, but these were only completed c1950, under BR's auspices.
By 1950, the only BR lines in London that were electrified were the Southern (on the former LSWR dc system), the LNWR local lines to Watford (which had to be to the LT 4-rail system), and the former GER local lines to Shenfield (1500Vdc ohle).
By the 1955 Modernisation Plan, ac technology had become more portable and all subsequent projects have been built to this standard, and some (e.g the GE lines) have been converted. This includes the longer-distance services out of Euston, the "dc lines" (now the Overground) being conveniently self-contained.

Both ends of Crossrail (Abbey Wood and Maidenhead/Reading) are south of the river too, and that will be OHLE.

Where goods trains are a significant part of the traffic OHLE was usually preferred because of the dangers of live rails to staff in marshalling yards. Some locomotives, such as the Southern's classes 70 and 71, and the NERs two Tyneside shunting engines, could take power from both above and below, so that they could operate on the 3rd rail fitted local lines and in marshalling yards where overhead cables were fitted.
www.semgonline.com/electric/pics/srly_cc1-3.jpg
www.railuk.info/gallery/diesel/pretops/26500_b.jpg

melikepie - talk of the "Electric Spine"seems to have slipped below the planning horizon in the last two years, presumably on cost grounds.

For Thameslink and the Milton Keynes service

It still seems to be going ahead

link

A few places with both - North Pole on the West London Line, Drayton Park on the Great Northern Moorgate branch and between Farringdon & City Thameslink, Camden to Euston slow lines and not sure if it has been replaced following recent/current upgrade platform 9 at Watford Junction (down fast south end of) that permitted units to transfer to/from Bletchley depot for maintenance.
Any other dual voltage sections?