Anyone know if there is an approved signal under NFHS mechanics for indicating that a team has taken its final timeout? What about NCAA? Other systems?

No NFHS signal. R is to notify Head Coach.

Both Head Coaches should be notified

Funny you should ask, a few weeks back i was working with a WH thats been officiating for 30 years. He faced the press box and made three downward movements with his arm (just like when kids back the gesture to truckers to blow their horn), signal the side of the team, and gestured the incomplete/declined signal. I asked him after the game and he said thats the signal to notify game administration that the team is out of timeouts for the half.

The three steam whistle tugs is an NCAA mechanic after a team as used its final timeout.
It is documented in the various CCA mechanics manuals. This signal does not replace notifying the coach. That is still done.

NFHS guys told me about the trucker horn signal. I only use it to tell the other guys on the crew that the team is out of TO's, but I always notify the coach on my side anyway when I'm the wing (which I usually am).

Plus, I just like going CHOO-CHOO!

There is no official NFHS signal for that situation. The HL & LJ should, during the game remind the HC of hie TOs left. After the third the WH informs the HC that he is out of TOs.

Our HS board has been using the three pull-down signals followed by three incomplete signals than pointing to the team on the field that has exhausted their TO's.

Plus, I just like going CHOO-CHOO!
 

That is fun. If you're the LJ and are facing the R when he's making the signals, you can synchronize it.

Wooo-wooo! chugga chugga chugga chugga....

No signal - R announces over microphone that the team has used its final time out.

What's a microphone?

A microphone?

It's what Elwood traded the original Bluesmobile for.