It is quite possible to create a truly mechanical computer. Babbage designed a working programmable computer (the Analytical Engine) but never properly built it. (He built a partial version, which worked, but had no memory. It did have a JMP command equivalent.)
It is entirely reasonable that, given the proper motivation, it could have been miniaturized by the Swiss watchmakers into something small enough for a robot...
Babbages was desinging a 5 ton monster with 20kB and <10Hz... my TS1000 was 4kB and 3.25MHz...
Given 20 years and decent funding, plus a half dozen swiss watchmakers, I'm certain that by 1930, the analytical engine could have been reduced to about 5L (and some 10-20kg), including a punchtape interface... but you're still looking at SLOW... possibly as high as 100Hz... 30,000 x slower than the TS 1000, 50x the size.
The theory was present by 1880... just not the funding.
http://en.wikipedia.org/wiki/Analytical_Engine
http://en.wikipedia.org/wiki/Difference_engine