There is a good chance that Jim MacArthur's machine (mentioned in the question) is in fact the first.
Here are the contenders I am aware of:
The Analytical Engine. As joeytwiddle points out, it has not yet been implemented.
Konrad Zuse's Z1. It had I/O, memory, and an arithmetic unit, but it did not have branching. So, not Turing complete.
Relay-based machines. The linked SE question lists a number of computers that, it says, were mainly relay-based. You could say these were not electronic, but rather electromechanical. They are full-fledged computers, but I am inclined not to count them as fully mechanical.
In 1978, Danny Hillis and Bryan Silverman designed and built a tic-tac-toe-playing machine made from Tinkertoys. But it was not fully programmable, being more an encoding of the full game tree of tic-tac-toe.
It should be possible to construct a machine that runs under steam power (or is fully belt-driven anyway with no internal electricity required), uses punch cards for bulk storage, and can do general computer tasks like calculation and text processing. But it would be a significant endeavor and would get a lot of publicity. The Analytical Engine is a bit like that. I think a project to make something even fancier would be an unlikely boondoggle.
All of that said, I was not familiar with MacArthur's machine until I read this question, so it's possible another hobbyist has beaten MarArthur to the punch.