Production wire wrapping is done using a semi automated wire wrap machine. These machines move the wirewrap bit the correct pin but the operator must trigger the wrap after aligning the bit on the pin which is why they were semi automated, an operator was still needed.

A picture of a Gardner Denver wire wrap machine from Litton advanced systems

WWM-380 Std. Logic Wire Wrap
This machine could wrap approximately 125 wires per hour of 24 to 30 AWG over a 24" x 40" area.