Use of the word "mechanical" to describe formal reasoning predates computers.

Here's the first sentence of Godel's 1931 On formally undecidable propositions...

"The development of mathematics in the direction of greater exactness has—as is well known—led to large tracts of it becoming formalized, so that proofs can be carried out according to a few mechanical rules."

Leibniz had discussed calculating machines (and even thought about binary arithmetic being the most appropriate implementation), so the general idea probably goes back quite far

Edit: Oh, I guess by "late 1930s" you're referring to Turing's 1936 paper where he defines Turing machines, rather than actual electronic computers. Still, understanding "formal" as "mechanical" predates it.