The Sundial Cannon of Åtvidaberg (2017)

Wouldn't this setup actually work though? At least if it's a proper sundial cannon.

Most components of that system are mechanical clocks, which do keep the time on their own, they just drift. The only exception is the gunner who fully relies on an external time signal.

So if that were all, the system would work for a while, but slowly drift and become ever more inaccurate, until it basically has nothing do do with the actual time anymore.

However, the gunner does not always operate the cannon: He only does on cloudy days. On sunny days, the cannon operates itself, using the time signal of the actual sun - which is then passed through the chain and let's the mechanical clocks resync to "sun time".

So in effect, the system's time signal comes from the sun. The rest is just an overly complicated way to "interpolate" the time if the sun is not visible or not at noon.

(Now I also wonder if someone ever build a "self-syncing" mechanical clock - e.g. for a clocktower - using this principle: Use a standard clockwork, but add some mechanism that resets the clock to noon when a focused beam of sunlight hits a certain point.)

The recoil both gives you kinetic energy and an impulse that is timed at noon. You could use it to drive some mechanism that prepares the cannon for the next day, e.g. refills gunpowder and adjusts the magnifying glass.

The latter could be done by having a gear with a cam that is driven 1/365 forward with each firing of the cannon and that encodes the position of the magnifying glass for each day of the year.

That only leaves you to design the exact shape of the cam. And deal with cloudy days and leap years...

Only when used in conjunction with a standardised 'mean-time' 24 hour clock. If you plot the position of the sun at the same mean clock time, then sometimes the sun will be ahead, sometimes behind.

So it's still local noon.