Other mistake: When Martha May is getting ready to put up her lights with the cannon, she pulls a charging handle, and starts firing. If you watch closely, you see a lit string of lights come out, but the spool that they're on isn't moving, or lit up. Then after all of the lights are out, it still fires several more times (watch the aiming indicator- it continues to flash with the sound).
Other mistake: When Lou and Cindy-Lou come home, Betty asks Cindy-Lou to unscrew the bulb in the refrigerator. A few moments later, Lou answers the phone, and repeats the question "Is my Sub-Zero Chillabrator running." Betty seems to have used an un-Who like term.
Answer: By machine.
The clock in "How the Grinch Stole Christmas" is designed for cinematic effect; a real-world counterpart would rely on precise gearing and engineering to allow three separate wheels to operate in close proximity without interfering with each other. The key would be in the gear ratios and the alignment of the gears to ensure smooth operation of each panel.