First, here's a picture of the insides:
The main component is a transformer with a 120VAC primary winding, and in this case, a centre-tapped 12.6V 3A secondary winding (which is not used). I bought it from Radio Shack when we still had those here in Canada.
I first removed the gold coloured outer shell from the transformer. Next, I cut out the pieces of the steel core at the bottom of the transformer between the three uprights, leaving the core in the shape of an upside-down Ш.
I then put back just the top half of the shell (as you can see in the photo), and epoxied the whole thing to the underside of a plastic project box lid (so with the lid on the box, the Ш is now right-side-up).
The secondary leads were cut short so that there's no danger of them touching one another.
The primary side was simply wired to a two prong electrical plug cable, with an AC-rated momentary push-button switch in one lead.
That's all there is to it.
To use it, hold the watch in one hand as far away from the device as possible. Push the button with the other hand, and bring the watch to within about half an inch of the surface of the box over the transformer, dial down. I usually keep my fingers between the watch and the box to ensure it doesn't get pulled out of my hand or get too close to the box. Wave the watch around slowly for a few seconds, and then move the watch as far away as possible before finally releasing the button.
A few warnings:
Don't press or release the button with the watch near the demagnetizer, or you will more than likely magnetize it.
Don't wear a ring on any fingers that you will bring close to the coil. The ring can become a very low voltage, high current, secondary winding of the transformer, wired in a short circuit around your finger.
And finally, don't wear a watch on the wrist of the hand you are pushing the button with, or you might magnetize that watch.