One problem I haven't found a mechanical solution for yet is how one could (simply) implement a state transition table - for a specific example say you have 9 states each mapping to one of 9 other states, and many-to one mappings are possible:

  1 -> 2
  2 -> 9
  3 -> 1
  4 -> 6
  5 -> 2
  6 -> 6
  7 -> 1
  8 -> 8
  9 -> 9

(This is the 3rd of 4 transition tables for an 8-state, 4-symbol Universal Turing Machine. These transitions apply if the 3rd symbol is read from tape at the current head position - with all 4 transition tables implemented you could select between them depending on the read symbol. 9 is the halt state.)

The mechanism should remain in one state and then go to the next as indicated by the table, repeatedly. How would you mechanically implement this? A face cam with many grooves perhaps, starting and ending at different angles? https://i.imgur.com/aNPBcdh.png - while always moving a follower from the center of the wheel through the groove to its edge, with something like a https://en.wikipedia.org/wiki/Chebyshev_lambda_linkage, so the wheel stops at the next angle representing the current state?

The fact that there does not seem to exist a simple answer for even this seems to partially explain why mechanical computers were quickly given up on.