Would demand for through passengers be that great? Even if the line was extended to Clapham Junction, the likelihood is that anyone on SWT who currently changes to the Northern line northbound at Waterloo would still find it quicker to continue to do so (Besides, even if some people do revise their journeys to change at CLJ instead, that's only going to increase numbers on the Northern line between CLJ and Waterloo - that would in itself have zero impact on passenger numbers north of Waterloo, which is where the line is at its busiest)
It's more likely that people arriving at CLJ on Southern trains will find it useful to change to the Northern line at Clapham Junction for journeys into the city, but even there I'd imagine for many destinations it'll still be easier to stay on the train to Victoria and then take the Victoria or District lines, so that doesn't look to me like bringing overwhelming additional numbers to the Northern line.
Extending the Northern line to CLJ seems very obviously useful in terms of improving connections for people whose final destination is in South London, or who want to get to the Northern line City branch, but I'd be surprised if added so many passengers north of Kennington as to overwhelm the Northern line.
Not only that, but aren't there plans afoot to increase the frequency of trains on the Northern line by fully separating the City and Charing Cross branches, removing the bottleneck at Camden Town?
1) Camden Town will need re-building before there is any chance of that happening, even exit / interchange only it can't handle passengers as it is, and non-stopping everything will only move the load to Euston, that can bearly cope as it is and will have no chance in any way when NLE2 comes into play and it pushes 200tph through the tube station and 64tph through Euston Square. Much more is needed before this kind of service level can be sustained with four escalators and a very poor interchange layout with single ended platforms and cramped duplication areas.
2) There currently isn't any space on the City Branch in terms of capacity for reaching the city, so even moving the city one change closer with CLJ - Kennington - Bank as a route it will tip Bank / Monument over the edge and result in dangerous levels of passengers not being able to clear from the platforms before the next train, so increasing tph isn't an option either. So moving passengers onto this route is far from 'loads of space'
3) Passengers travelling from CLJ on SWT to the West End have a choice of the Bakerloo and Northern Lines currently, the Bakerloo has a reasonable amount of capacity available where additional space can be gained by upgrade as part of the DTP. Moving the Northern Line to Clapham Junction would result in almost all suburban SWT passenger changing here rather than Waterloo and revising their travel plans to use Tottenham Court Road or Leicster Square rather than Oxford Circus or Piccadilly Circus, again, lack of capacity at Leicster Square is an issue, let alone shoving all these passengers onto one set of trains that will be capped at around 24tph due to line capacity and space at other stations.
4) All this also depends on well, having trains...(!) Something the Northern Line is currently lacking enough of.
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(off topic somewhat) where would that diverge?
I'd diverge at either Pimlico or Vaxhaull or possibly Victoria, not Stockwell.
Although personally I'd more support platform 5 at Victoria, using 4 for terminating trains into 22/23rd and fast turnarounds, 3 for Northbounds, 4 for terminating or occational northbounds (install stepping back) and 5 for all southbound services, encouraging pax to change at Green Park if continuing or digging out the rings to have 4 as a double faced platform.
