Agree on trying to base ANYTHING on History channel film choices...
The various Ford factories did, however, build a LOT of flatheads for all sorts of uses in WWII, a few for cars, many for trucks, some in armored vehicles, pump and welder motors, a few actually flew...look around here for starters...search function.
Principal engine Ford builders for WWII in my guess of order of size were Ford USA, Ford Canada (supplied all Imperial armies), Ford England, and Ford Germany. Russia was building various Ford A-B based vehicles (Model B's also served in British and German armies), but the Russian plants were their own brands, using machinery and plant once purchased from Ford USA.
There are good Ford at War books around from USA, Germany, Canadian, and British viewpoints.
On the civilian front...vast numbers of engines were also remanufactured during the war both for mi and essential civilain uses.