The save throw occurs when you launch Order. Targets that fail to save may follow the command on their next turn.
According to Players Manual, the effects of a spell occur when it is cast.
Many spells specify that a target can make a save roll to avoid some or all of the effects of a spell. The spell specifies the capacity used by the target for the backup and what happens if it succeeds or fails.
When a creature is prompted to make a save roll in response to an event (such as a spell), it starts casting immediately.
A saving throw, also called a backup, is an attempt to resist a spell, trap, poison, disease, or similar threat. You do not normally decide to make a backup roll; you have to create one because your character or monster is at risk.
The result of a successful or failed backup is also detailed in the effect that allows the backup. Generally, a successful backup means that a creature does not suffer any harm, or reduced harm, from an effect.
The majesty Coat of Glamor Bard technique allows them to launch the Order fate, which, if read clearly, may seem a little fuzzy on the exact moment when the save throw of Wisdom takes place:
The target must make a successful Wisdom save roll or follow the command on their next turn.
The save roll is part of the spell and you cast the command as part of the spell. If the target creature fails, on its next turn, it will follow the command.