"The target begins a comic dance..."
Choose one creature that you can see within range. The target begins a comic dance in place: shuffling, tapping its feet, and capering for the duration. Creatures that can't be charmed are immune to this spell.
While you are casting the spell, you choose the target. As soon as casting the spell is complete, the target "begins a comic dance" (unless it is immune). The rest of the spell description is about the effects of this dance.
Some of the effects of its dancing happen immediately:
A dancing creature...has disadvantage on Dexterity saving throws...While the target is affected by this spell, other creatures have advantage on attack rolls against it.
Other effects 'begin' on the target's next turn. It is not that these effects are delayed, merely that the things they affect are not relevant until the target's turn, when it first has movement and the ability to take actions:
A dancing creature must use all its movement to dance without leaving its space and has disadvantage on...attack rolls...As an action, a dancing creature makes a Wisdom saving throw to regain control of itself. On a successful save, the spell ends.
It is a bit confusing that the immediate effects and the effects that later happen on the target's turn are interspersed in the text, but since we know that the target immediately "begins a comic dance", and since all the effects of the spell are consequences of this dance, we can apply each of the effects at the first moment they become relevant.
Because the target is dancing for the entire duration of the spell, some effects that normally would not happen until the target's turn may be "moved up" before its turn, depending on circumstances. For example, if another creature left the target's reach after the spell was cast but before the target's next turn, the target would be at disadvantage to hit should it choose to make an opportunity attack. Another case would be if the target was granted movement before its next turn, such as when a spell effect used its reaction to force it to move. In this case, the "must use all its movement to dance" would take place as soon as the other feature granted it movement. Cf. Does the target of Otto's Irresistible Dance followed by Dissonant Whispers (failed saving throw) move as a reaction or just keep dancing in place?
The OP asks:
if other party members want to fireball the creature or attack it before the creature's initiative count rolls around, will they already benefit from the effect?
As explained above, yes! The spell is particularly powerful when the caster can grant advantage to their entire team on a target before the target can even get a save against the spell. Compare this to, for example, web, where the targets are not restrained until their own turns, at which point they have a chance to save, so unless they fail there is no direct benefit to the web-caster's team.
However, automatic full-team advantage occurs only when the caster's and the target's initiatives happen to 'bookend' the rest of the caster's team's initiatives, and bear in mind that this is a sixth level spell that targets a single creature.
I believe that it remains an open question of whether a target could, on its first turn, elect to use its action to end the spell before it moved and, if successful, could then use its movement as normal, or whether all of its movement is immediately used up as soon as its turn starts and before it has the chance to make the save.