What is the role of this Zener diode in a LED string? (Old traffic light restoration)

Little I think. The desired controller section would only consist of an Arduino Uno R3 executing a single light phase routine* via a MOSFET module wired to the LED PCB. The 3 FETs will be operated by 5V signal output from the Uno pins. At 1 pin per color this makes 3 times 40mA = 120mA at the most I would say? In practice probably less.

*) red > green > yellow > red > .... to infinity. Super basic.

That would ideally be the case. I'd like to have something like this so the whole traffic light has just 1 power plug, which in turn takes care of powering all internals (Uno, MOSFET, LED PCB):

But this would mean I need a 48V capable step-down buck converter and a 48V capable MOSFET board as that is the native voltage the LED PCB requires (and won't take less). However both such buck and FET I cannot find. I do have a switching 48V 800mA wall-socket power supply.

If I'm not mistaking, is that what you are describing as well?

OK, How about some ugly. Get two 110 volt to 24 volt step down transformers. The primaries can be connected in series for 220 volt input, or parallel for 110 volt input. The outputs can be connected in series for 48 volt AC power. If you want it all in one transformer the DST3-48 may fit your needs. Arduino drives relays or opt- isolators to drive the LEDs.

My idea was to use a linear regulator such as LM317 to drop the supply voltage to some lower value your buck converter will handle.
I don't know why you need a MOSFET module. There is plenty of MOSFETS (and BJTs and Darlingtons and relays) that will handle the required current and voltage.
EDIT: and you surely want low side switching with a N-channel/NPN device.

Arguably, these modules - the version using the FR120N - would correctly switch your 48 V (DC) to the lights.

A foolish question: Are you making a real traffic light for roads or a traffic light system just for small project?

There are Many MOSFET capable of handling 48V and above. E.g. - IRF740 capable of handling 400V Drain to source voltage and upto 10A of Current. It all depends on what is available to you. (It is available on the Tinytronics website of which you posted links in post#38.

I think you can use an external 9V PSU for powering the UNO, because 48V Buck converters are available but, I can't guarantee that if it can handle 48V or not.

Regards.

A thought on the Zenner diodes.

If an LED was to fail and opens the voltage on the other three strings would rise. Thus increasing the possibility of a failure in the remaining three strings. If an LED in one of the remaining three strings fails open, the voltage would again increase on the remaining two strings. If an LED in the remaining two strings . . .

For reliability lets keep the surviving strings operational. In life/safety issues reliability is job one.

Only if a ridiculously small transformer would be used. This is not a concern in practice given the current levels we're talking about here.