I have posted on this before, but then I have done much since. Many issues for London Underground and London Overground are now covered in my suggestions maps.
Diagrammatic Map excluding Crossrail 2 and Barking Riverside
Diagrammatic Map adding Crossrail 2 and Barking Riverside
Geographical Map excluding Crossrail 2 and Barking Riverside
Geographical Map adding Crossrail 2 and Barking Riverside
The London Overground is turned into Chords so that routes are recognisable. They are named to be memorable and related. A Chord is an adopted name for an outer non-central serving railway, but for simplicity the Overground can call all its routes Chords. They offer travel choices: perhaps longer journeys but less crushed at busy times, different routes to the same place, negotiating pay zones better. In addition to Chords are the functional routes like for airports or crossing the river - all services would be part of the same ticketing, including the boats and the buslinks.
There are four stations on Holloway Road so Holloway Road is renamed Holloway. Caledonian Road and Barnsbury is renamed The Cally, as it says on the bridge. There is now only one Bethnal Green so the other is renamed Weavers Fields. The decision to call a station Battersea Power Station when it isn't one is reversed in my suggestion, where the line is extended to Clapham Junction one way and connects with the (missed) Victoria line the other via a new station of Lambeth South. That would be expensive, but the extension is for little real benefit and having to go to Kennington before turning north is ridiculous.
The Northern line as was is broken into its two logical lines: Unity line and Cricket line, according to my renamings. The District line is reduced so that the Tennis line runs between Wimbledon and Edgware. The Metropolitan line is less unwieldy by taking advantage of the Overground and Buslink so removing its Uxbridge spur and relocating West Harrow.
The Victoria line is extended to the north beyond its sidings to Angel Road and to the south beyond Herne Hill to Tulse Hill (for better connectivity). The Bakerloo line is extended but only to Lewisham - the Chords do the rest. The Cricket line, as renamed, goes into the Morden sidings and across the road to join the railway there and thus extend the Underground route to Sutton. The Tramlink is extended to South Wimbledon, across to Bromley - as its council prefers, rather than the Hayes line (Wolf Chord) becoming the Bakerloo - and the tracks to New Addington continue down along/ alongside the grassy areas and roads out across to Biggin Hill Airport and its settlement. This is why a Gatwick Express service must stop at East Croydon.
York Road is reopened as York Way and Maiden Lane is reopened too, providing a walking connection. It doesn't take masses of money to create some Overground connections. Brixton is awkward but not impractical.
What was the Northern Heights Railway, now Mill Chord, might cost some money, but the track bed is there - use of overlapping and single tracks can work with sufficient passing loops and double lines.
Buslinks link spurs and outer stations where useful. Goods lines are brought into passenger use, so better goods lines outside London are necessary. A podlink is driverless programmable vehicles on guided roads to help (in the case of Brent Cross) with the shopping. There are no guided buses.
One would like to see anything near this kind of investment in the north, but the capital city was starved of investment for a long time and needs it to get people moved about more effectively.
Some issues like Charing Cross being, really, two stations and Embankment as actually for Charing Cross are better left to sleep on. Dedicated walkways do the connecting and we now have walking down streets elsewhere. After all, if Crossrail 2 is built there will be one station called Euston Kings Cross St. Pancras International, and include Euston Road. I limit connections to 500 yards but even that is too far for many people. Such outdoor routes need to provide shelters and even covering. So many platforms are outdoors, but they provide shelters.
As for airports for London: there's Heathrow, Luton, Stansted, Gatwick, City and don't forget Biggin Hill.