I play a Glamour bard with the Mantle of Majesty feature (XGtE, p. 14). The feature lets me cast command on a creature as a bonus action.

My DM and I currently are arguing about when the enemy gets to make the save against command. I believe it's right when I cast the spell, but he believes it's when it's the enemy's turn. I haven't been able to find anything about the timing of the saving throw.

The saving throw happens when Command targets a creature. Creatures who fail the save can follow the command on their next turn.

According to the Players Handbook, the effects of a spell happen when the spell is cast. If the spell prompts targeted creatures to make a saving throw, then the affected creatures should do so.

Saving Throws

Many spells specify that a target can make a saving throw to avoid some or all of a spell’s effects. The spell specifies the ability that the target uses for the save and what happens on a success or failure.

When a creature is prompted to make a saving throw in response to some event (such as a spell), they make the roll immediately. Otherwise, the spell effects wouldn't be resolved, and the outcome of the spell would be undetermined. Therefore, creatures make the saving throw when they are targeted or affected by the spell.

A saving throw—also called a save—represents an attempt to resist a spell, a trap, a poison, a disease, or a similar threat. You don’t normally decide to make a saving throw; you are forced to make one because your character or monster is at risk of harm.

[...]

The result of a successful or failed saving throw is also detailed in the effect that allows the save. Usually, a successful save means that a creature suffers no harm, or reduced harm, from an effect.

The Glamour bard's Mantle of Majesty ability lets them cast the Command spell, which, due to awkward punctuation, may invite disagreement about when the Wisdom saving throw occurs:

The target must succeed on a Wisdom saving throw or follow the command on its next turn.

While the consequences of the spell (i.e., the target acting according to the command) don't happen right away, someone still needs to determine whether or not the target was affected. You issue the command as part of the spell, and so the saving throw should happen as part of the spell. If the targeted creature fails, then on their next turn, they will follow the command.

Saving throw is rolled when the spell is cast.

The command is followed on the target's turn (if save fails)

From the spell description

Command

... target must succeed on a Wisdom saving throw or follow the command
on its next turn. {emphasis mine}

You roll the save right away, but whatever it is that you want the target to do will happen on its next turn.

A saving throw—also called a save—represents an attempt to resist a spell, a trap, a poison, a disease, or a similar threat. You don’t normally decide to make a saving throw; you are forced to make one because your character or monster is at risk of harm. ... You don’t normally decide to make a saving throw; you are forced to make one because your character or monster is at risk of harm. (Basic Rules, p. 65)

Command's description doesn't call for the target to obey your command as a Reaction (on your turn); it would say so if that were the case.

The target is limited to doing something as an Action, on its turn, if it fails the save.

The result of a successful or failed saving throw is also detailed in
the effect that allows the save. Usually, a successful save means that
a creature suffers no harm, or reduced harm, from an effect. (Basic Rules, p. 65)

A note on "at the table consistency"

Some spells call for a save right away, when they do damage. For example, shatter. The wizard casts the spell, the damage is done, or have damage with a save, right then, as a matter of resolving combat. It is worthwhile to be consistent (so that DMs and players don't forget things) to roll the saving throw when the spell is cast. The fact that some spells call for a saving throw on each successive turn - for example hold person - is generally spelled out in the text.

From a consistency and "keep play moving" perspective, not making the saving throw when the spell is cast would seem to be the outlier.

Ambiguous

Specific beats general, so the primary source must be the spell text.

The spell text reads:

[...] target must succeed on a Wisdom saving throw or follow the command on its next turn.

This can be read two ways. Both are perfectly reasonable.

The first reading is:

• the target must succeed on a Wisdom saving throw (at an unspecified time), or follow the command on its next turn.

And, by default, you save against spells when they occur. So you save immediately.

The second reading is:

• (the target must succeed on a Wisdom saving throw or follow the command) on its next turn.

which is a totally valid way to parse that phrase. The phrase is worded ambiguously.

In this second reading, the saving throw occurs on the next turn after you cast the spell.

Spells can cause saving throws after the moment they are cast; hold person, for example.

Managing the Ambiguity

For this spell, the wording is ambiguous if the saving throw should occur immediately, or on its next turn. So this is a clear case of "DM's call".

Both make sense logically; either the target resists the command spell now when it is cast, or the target has a chance to resist it when it has an effect. Neither interpretation causes any serious balance issues.

Effects on the Game

That is not to say there are no concrete effects on the game.

For example, if the target has a Paladin ally, and that Paladin ally moves towards/away from the target between the point of casting and the target's next turn. But these are relatively small in the grand scheme of things.

In addition, the DM might inform the caster of the failed save if it happens early, so that might give them some earlier information.

The General Rule:

You don’t normally decide to make a saving throw; you are forced to make one because your character or monster is at risk of harm.

there is no harm from the Command spell until your next turn, so there isn't even a general case being overriden here. There is a general pattern that spells cause effects to be saved against immediately, but there doesn't appear to be a rule that states spells in general are saved against immediately.

There are many spells that aren't saved against immediately. They include hold person and delayed blast fireball. In both cases the save is mentioned in a context of a later turn. Those cases are clearer, as the sentence the saving throw is mentioned in isn't ambiguous, than this spell. But there is plenty of precident.

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.