Gary:
The Wolcott lathe is a classic "cone drive" or "cone head" engine lathe. You are lucky to find a lathe of this age and type intact and unmolested. It is a real old workhorse.
As for it being used on any production work at Ford, whether for automotive or defense work, I kind of doubt it. The lathe is not a production machine tool. Production machine tools are typically built to do one or several types of operations repetively and are built to "run 24/7". An example of a production lathe would be a turret lathe, rather than the engine lathe that you have. With plain bearings (such as your Wolcott lathe has) and no features for repetitive/production machining, your lathe was most likely used at Ford in a different way. Namely, it could have been in a maintenance shop in some place like the powerplant or one of the auto plants. The automotive plants required vast amounts of support in the form of toolrooms to make/maintain the tooling (jigs, fixtures, dies, specialized cutting tools, etc) used in the production of the automobiles or defense work. The toolroom machine tools were usually highly precise machine tools, sometimes made to a higher standard of accuracy than the regular machine tools. Then, with the sheer volume of machinery and physical plant in the automotive industry, there had to have been maintenance shops. Ford ran miles of railroad in their plants (such as the Rouge), had their own blast furnaces, steel mills, ore boats, ore unloading facilities, and on it went. With countless pieces of machinery, equipment, and plant systems to be maintained, my thinking is your lathe likely came out of one of the in-plant maintenance shops.
If it WERE ever used in production, you would not be finding it in the condition you did. It would likely have gone to the razor blades long ago as it would have been so worn and beat that it was beyond any recall. Even in the Model T days, Ford was investing in production machine tools, and I recall reading how Bullard built some semi-automatic vertical turret lathes for machining the flywheels. Ford was quite please when they found that the Bullards could machine a flywheel in a matter of maybe a minute and a half, if that long. Similarly, any production lathes on the "line" would have been turret lathes or some other variant such as LeBlond made- usually with no lead screw, a clutch/brake on the spindle so the operator could get jobs in and out quickly, and means for setting a bunch of "stops" for repetitive operations. Any machine tools used in regular production in the automotive and defense plants had to earn their keep, and it meant some "time study guys" with stop watches and clip boards were watching every move the operators made and trying to shave even a few seconds off each production machining job. No one would "make the time" on a manual engine lathe like your Wolcott lathe. On the other hand, it DID play its own part in keeping Ford going, by supporting the plant in either a toolroom or maintenance shop.
Another location the old Wolcott lathe could have been located in might have been the Henry Ford technical/trade school. Henry Ford was a great believer in teaching young men (sorry if not PC, but those were the times) the machinist trade. He saw to it a fine and well equipped school or schools were set up within the Ford Motor Company to train young men to become machinists, toolmakers, and other technical professions such as draftsmen and tool designers. Since your Wolcott lathe is not "beat on" as many lathes of that same era would be, and seems to have seen some use but well cared for, my guess is it might have even come out of the Ford trade schools.
It was likely tucked away in some quieter area of the Ford plants, well cared for, and not used long enough or hard enough to show hard wear and damage. It is also unmodified. Many old lathes of this era would have had some sort of brackets or supports cobbed onto them for a motor drive. Many lathes of this era, having been pushed aside when the move from lineshaft driven machine tools occurred would have been through a few moves and a few owners. In the process, it was not uncommon for the old lathes to sustain damage and wind up with some of the handwheels and levers broken and brazed back together. It is also not uncommon to see old lathes of this era covered in umpteen coats of paint slathered on with a wide brush. Your lathe looks like it may well be in its original paint. Again, a production machine tool would have the paint worn off in areas from coolant and chips. Look at the LH corner of the compound (top slide) where the toolpost is mounted. If it is chewed up, it means the lathe was used by indifferent operators who had occasional minor "crashes"- running the compound into the chuck jaws. Students did this to some degree as they learned to run lathes. A few nicks on the corner of the compound are common, but if it looks like someone fed the corner of the compound into the chuck jaws and really took off some meat, that tells the tale of a lathe that was not well cared for.
The bed of the lathe is some type of cast iron or "semi steel" (cast iron mixed with scrap steel). Not a hardened bed, so would have been hand scraped when the lathe was built. If you can see the "fish scale" or "frosting" from the finish scraping along the bedways over the run of the bed, the lathe would be in incredibly good condition, or had been rescraped and seen little service afterwards.
Ford had what approached an infinite number of possible non-production applications for your lathe, and it would have supported anything from basic plant maintenance to toolroom work or training. Your lathe did its part, but not in the way you might have first imagined. BTW: I'd move the tailstock in so it is sitting fully on the bedways before it lands on your foot.