As I understand it, the tube has to be closed overnight because of
maintenance work. The NYC subway has 4 tunnel lines so they can be open
all night.

What about a limited service using just one track? You'd have one train
running back and forth between 2 to 3 stations. Take the victoria for
example, one train would run Brixton - Stockwell - Vauxhall - Pimlico -
Victoria and back again, giving a frequency of one train every 20 minutes
in each direction, then a train would run victoria to kings cross, again a
20 minutes service, kings cross - seven sisters, seven sisters -
waltahstow.

As only one tunnel would be used, the other could be maintained. Until 4AM
the trains could be timetabled to allow leaving the center easilly,
minimising connections - on the victoria line the victoria - kings x train
would arrive, then leave, then a minute later the kx - seven sisters train
would arrive. After 4AM the flow would be reversed allowing access to the
city.

If the service was popular enough, you could have trains running every 10
minutes, so brixton-vauxhall, vauxhall-victoria, vic-oxcirc ox-kingsx,
kingsx-finsbury, fins-tottenham hale, th - walthamstow
Central could be ealing broadway - north acton - white city - notting hill
- bond street - holburn - liverpool street - stratford - leytonstone
Piccadilly could be finsbury - kingsx - holburn - green park - glocester
road - Hammersmith

Well, you get the idea. Going against the traffic flow would take a long
time, but going with the flow should only take an extra 10 minutes per
hour. I'm guessing outside the tunnels it's easy to switch trains from one
track to another allowing trains to pass each other at key points. With
careful timetabling and planning, routine maintenence could be done
while a limited service would still be operational.

Is there a demand for all night services?