Rewinding a pickup by hand is an impossibly task in that it takes forever to unwind and rewind every coil. We're talking a couple of weeks of dedicated work here...
Unfortunatelly I was in a situation were I had to unwind and rewind a pickup that went dead.

Fortunatelly I thought of the Lego kits I have kept from my childhood (that wasn't so long ago) . I spent a couple of hours and eventually made a machine that would unwind the pickup unattended and would help me rewind the pickup quite fast. (see vid)

 
All it took was a multi-voltage adapter, a 9V lego electric motor, a couple of lego axls and gears, a big lego rim as temporary bobbin for the wire and some lego bricks.
Feeding the motor with only 3V made it turn slowly without enough power to break the wire in case the latter got stuck for any reason. This allowed me to let it work safely on its own for a some hours until the spot the wire was busted came to the surface.
After joining the broken ends and restoring continuity through the pickup, I switched the position of the bobbins and began rewinding the pickup. This part of the task I had to supervise and lead the wire onto the pickup bobbin. However I raised the voltage, accelerating the motor and speeding up the process.
 
I shot a short video of the winder in action: