All the effects operated via simple pneumatic pistons. Since it was all just done by means of compressed air back then, gags would pop up or swing into view just as quickly as they would in any other pretzel-style dark ride. You can actually see the motorcycle cop from the original Mr. Toad in action in the movie
Forty Pounds of Trouble from 1962. He had a jointed leg that he would kick up, providing the illusion of starting his motorcycle, and the movement was very swift. (I've uploaded the dark ride scene here, as the old YouTube uploads of the Disneyland sequence from the movie were blocked a while ago:
https://drive.google.com/drive/folders/1YX9ee_yEVgKNcwwzvyl4y_oyXvdYhpiE)
The attached image is a bit from an interview with Bob Gurr, the guy who designed all of the animated gags for the original Fantasyland dark rides, regarding the original Alice attraction. This is from issue #31 of
The "E" Ticket, and it goes into detail about how the gags in the ride operated. And you most certainly can hear the mechanical workings in the audio recording, by the way. If you listen closely, you can make out all sorts of muffled little pneumatic snaps, as well as Alice's narrations and other audio reels bleeding in through other scenes.