As for the Apollo flight computer, a very limited orbit-tracking version might have been possible but integrating error would have made it deeply suspect over such a long time period I think. In terms of all the other things the Apollo computer did in terms of attitude control and timing the firing of thrusters correctly, I doubt you could make a one cubic foot mechanical or electromechanical computer do that.
I'm not suggesting that a mechanical computer could have replaced the Apollo flight computer. But if improvements in pre-calculated tables allowed ballistics and even rocketry to develop a little faster, mechanical computers might have come in handy for pre-Apollo rocket launches. What's the minimum computer functionality required to put a man into space? On the moon? And maybe some of the computing work could have been shifted away from the vehicle to a dedicated Flight Computations building on the ground.