The Chuul makes a single-action Multiattack; the tentacle use is part of that action.
From the Sage Advice Compendium V.2.3 (11):
Is the grappling rule in the Player’s Handbook usable by a handless creature?
The grappling rule (PH, 195) was written for a
grappler with at least one hand, but a DM can easily adapt the rule
for a handless creature that has a bite or an appendage, such as a
tentacle, that could reasonably seize someone. A wolf, for example,
could plausibly try to seize a person with its bite, and the animal
wouldn’t be able to use its bite attack as long as it held onto the
person. Keep in mind that the grappling rule in the Player’s
Handbook requires the Attack action, so a creature must take that
action—rather than Multiattack or another action in the creature’s
stat block—when it uses that rule. A monster, such as a roper, that
has a special grappling attack doesn’t follow that rule when using its
The special grappling attack of the Chuul is the Multiattack with pincers and its tentacles. If the condition is met it does not have to spend an extra action. It does not follow the PHB's grappling rules.
Multiattack. The chuul makes two pincer attacks. If the chuul is grappling a creature, the chuul can also use its tentacles once.
The creature can choose to use its tentacle because it has a valid target at that point and the condition is met. The Multiattack is a single action.
Also see MM (10) on Actions, more specifically MM (11) on Multiattack and Grapple Rules for Monsters.
We also have some insight into design intent, albeit unofficial, in this tweet sequence and in particular the answer by Jeremy Crawford, who says:
The DM decides the order of a creature's attacks using Multiattack,
unless the creature's stat block mandates a particular order.
This ruling is supported by Simultaneous Effects (XGtE, chapter 2)
Most effects in the game happen in succession, following an order set
by the rules or the DM. In rare cases, effects can happen at the same
time, especially at the start or end of a creature’s turn. If two or
more things happen at the same time on a character or monster’s turn,
the person at the game table — whether player or DM — who controls
that creature decides the order in which those things happen. For
example, if two effects occur at the end of a player character’s turn,
the player decides which of the two effects happens first.
So what are effects?
DMG errata version 2.0 (1):
Combining Game Effects (p. 252). This is a new subsection at the end
of the “Combat” section:
Different game features can affect a target at the same time. But when
two or more game features have the same name, only the effects of one
of them—the most potent one—apply while the durations of the effects
overlap. For example, if a target is ignited by a fire elemental’s
Fire Form trait, the ongoing fire damage doesn’t increase if the
burning target is subjected to that trait again. Game features include
spells, class features, feats, racial traits, monster abilities, and
magic items. See the related rule in the “Combining Magical Effects”
section of chapter 10 in the Player’s Handbook.
So we can safely gather that monster abilities are indeed effects and that either way the controller of the creature decides the order of the effect. In this case, it is most likely the DM who may choose either way.