There are many combinations and procedures in assembling pickups. Each manufacturer has their own technique for making pickups. For example, some companies magnetize the magnets after they are assembled and some magnetize before they are assembled. I do both as it can change the path of the magnetic field it has to take and I also do single magnet calibration before the pickups are assembled. Some sample procedures for a single coil pickup.

Material punched to desired shape and punched for number of poles and pole piece dimension.
Magnets are prepared for desired cosmetic appearance and pressed into tooling that holds the top and bottom flatwork in place.
Bobbins are made clean of burrs and surface is prepared for winding.
Bobbins are wound with a specific grade of magnet wire to the desired number of turns, winding pitch, tension, winding direction and magnetized North or South magnetic polarity.
Bobbins are connected to desired hookup wires and coils are tested before potting in a specific solution to help eliminate unwanted microphonic feedback.