Prior to the 30's the tube didn't go far from London, except the Metropolitan, which had main line aspirations.
Then there was the unemployment relief plan and the tube was extended north by taking over or building alongside existing steam worked railways. Very little new right of way was created. The Southern regarded commuters as bread and butter. North of the river commuters were regarded as something of a nuisance. (Liverpool St and Fenchurch St lines were an exception)
Take a look at the UERL map of 1908 here and compare it with that of 'London' at the time
here. The UERL map was issued just 20 years after the LCC was formed and it shows that most of the lines (including the two south of the Thames) were providing services into the centre of London as fast as capital was available.
Fast forward to 1933, - the political boundary of London was still that of 1889, but it was becoming clear that the demand for travel into the centre was rapidly growing outside that area, hence the 'New Works' expansion utilising existing surface lines of the big four that in many cases were unsuitable for increasing commuting.
Roll forward to 1965 when the administrative area of London was expanded for =m the LCC boundary to the current GLC/GLA boundary. By then the New works had extended the original lines to reach out to and beyond the GLC border.
So in terms of coverage, the, now, TfL Underground railway cover most part of the region north of the Thames within the M25, (south of the river still being served mainly by the 3rd rail services of National Rail).
London, as a Metropolitan region is still expanding and a combination of TfL and National Rail services is still providing as far as is affordable, metro services. Crossrail is the latest to add to cross region connectivity to the centre, - maybe Crossrail 2 will be the next.