The save is rolled when the spell is cast.
At first glance, the wording seems ambiguous about the timing of the saving throw:
The target must succeed on a Wisdom saving throw or follow the command on its next turn.
In theory, one could read "on its next turn" as applying to only the effect or to both the effect and saving throw. However, with a little work, we can show that this timing cannot be interpreted as the trigger for a saving throw. Therefore the timing must apply only to the effect, not the saving throw.
First, if we look at other spells that require targets to make saving throws on their own turns, we notice a pattern: the spell is always very specific about exactly when the saving throw occurs. For example, the text of grease says:
A creature that enters the area or ends its turn there must also succeed on a Dexterity saving throw or fall prone.
Maelstrom:
...any creature that starts its turn there must succeed on a Strength saving throw or take 6d6 bludgeoning damage and be pulled 10 feet toward the center.
Moonbeam:
When a creature enters the spell's area for the first time on a turn or starts its turn there... it must make a Constitution saving throw. It takes 2d10 radiant damage on a failed save, or half as much damage on a successful one.
Compulsion:
After moving in this way, it can make another Wisdom saving throw to try to end the effect.
The point of all of these examples is that any spell which demands a saving throw at any time other than immediately at the start of the spell's duration says so specifically, and furthermore it always gives a specific trigger for that saving throw. In particular, if a creature has to make a saving throw on their own turn, the spell always specifies exactly when during their turn the save occurs: either at the start of their turn, the end of their turn, or when a certain event happens, such as moving into an affected area. The other pattern to note is that whenever a specific trigger is given for a saving throw, the information is always presented in the same order: first the trigger, then the saving throw, and finally the effect.
In contrast, the text of command does not follow this pattern. It specifies the saving throw first, then the effect, and then the timing. And compared to the above examples, the timing of "on its next turn" is not sufficiently specific. It does not give a precise trigger for when the saving throw occurs. (Of course, the obvious timing would be start of the target's turn, but it doesn't say that.) On the other hand, this timing does make perfect sense when applied to the effect: it says that the target spends its entire turn following the command. Hence we must conclude that the timing given applies to the effect, not the saving throw. Since no timing is specified for the saving throw, we fall back to the "default": the saving throw happens when the spell is cast.
Based on the above examples, we can imagine what the spell might say if the intent was for the target to roll the save on its own turn:
At the start of the target's next turn, it must succeed on a Wisdom saving throw or follow the command on that turn.