Scripts › library. Edit a script that can be scheduled, tick run it on a schedule, and change the timezone box.
Open Scripts › library. Set the kind to Find out what is wrong and press edit on Who is signed in. Tick run it on a schedule. The time is 02:00 and every day is ticked. The timezone box copies this console's clock. This picture has Settings › display on Australia/Melbourne, so the box says Australia/Melbourne. Yours says whatever that menu is set to. Show times in another timezone is that menu. It does not change a schedule you have already saved.
Replace the timezone with Australia/Perth. The time stays 02:00. After save changes, the script runs at 02:00 in Perth on the days that are ticked, on every machine it covers that is online then. A machine that is off is skipped. This shot stops before save changes, and the last step cancels, so Who is signed in stays unscheduled. The nightly pass does not move with this box — that clock is Run the nightly pass in another timezone.
Turning the schedule on is Run a script on a schedule. Leaving the weekend off is Run a script on weekdays. The console clock itself is Show times in another timezone. The nightly pass is Run the nightly pass in another timezone.