I really can’t remember what we called mechanicals (mechs) in Britain. Probably paste-up or artwork boards. Of course the heyday of mechanicals was the 60s and 70s: prior to that they weren’t called for as hot metal type had its own workflow, and later on as typesetting systems became more powerful and could handle page make up themselves, they became redundant.

In the early days of offset printing we were all fixated on the image we would use to set in front of the camera so it could be shot, converted to a negative, stripped up into flats and used to make plates. Straightforward books might come from the hot metal composing room already paginated, but as we sought to save money by using other equipment (e.g. Selectric typewriters) to create the type, the services of a compositor had to be replaced by you, the paste-up artist. Mechs were basically a paste-up of the type and images to appear on each page — lots of little bits of paper stuck to a board. Designers would get out their scissors, X-acto knives, tweezers and paste pots and assemble the pages. The text for the book would come from a typesetting machine as repros (reproduction proofs) and would be cut up and pasted down onto illustration board. In making up a page you might have one bit of paper carrying the running head which you pasted down at the top of the page. If you were lucky the boards would be preprinted in non-reproducing blue showing a grid where all the elements would go. If you were less lucky, you’d get to draw this up for yourself. Apply paste the back of the running head and position it centered at the top, then press it down, viewing it through one of these plastic rulers with a grid pattern so you could get it straight, then move on to the text. If there was a mathematical equation in the middle of the page, you’d probably have to position by eye 10-20 little bits of paper, making sure they were all properly stuck down, as well as aligned correctly. A piece of line art would be pasted down in position. A halftone would be left as a blank, to be shot and cut in at the printers in negative. Another few lines of text, and then you could paste up the folio at the bottom, then on to page 2. In some old books you can see the tell-tale signs of a hurried paste-up artist — a crooked folio, or even a missing one where the paste gave up and the folio fell off before the mech got to the camera.

The Oxford English Dictionary defines mechanical (after all the more obvious definitions) as

 3. Printing. A completed assembly of artwork and copy. Also in extended use.

1967   V. Strauss Printing Industry xi. 744/2   At its simplest, the mechanical is a piece of artists’ illustration board, somewhat larger than the final size of the printed piece. To this board are attached, by cementing or pasting.., a number of line images, all in the same focus and, of course, of inspected quality. This board bears, furthermore, all notations that will enable it to serve as the blueprint of the job.
1967   Britannica Bk. of Year (U.S.) 66   Many regulations, particularly in relation to TV, prevented advertisers from using international campaigns if the basic mechanicals—artwork, films, and so on—were not produced in Italy by Italians.
1973   Publishers Weekly 12 Mar. 38   The layout [of an advertisement] was changed at the last minute, and the mechanical bearing [the publisher] Quadrangle’s name either was not replaced, or it fell off.
I’ve found one earlier reference, from The Bookman’s Glossary, 4th edition 1961 R.R.Bowker Co.

“Meticulously prepared layout for engraver or printer, showing exact placement of every element, and carrying actual or simulated type and artwork”.
Notice that the 1961 definition makes the mechanical something used for preparing art for the engraver, as well as preparatory work for printing. I think that gives us the origin of the word mechanical as applied to book manufacturing. It was probably a term used by engraving companies, and was extended to cover the thing described above.