[RPG] How to a character escape after being engulfed by a shambling mound

Your players have a lot of options.

From the Grappled condition:

• The condition ends if the grappler is incapacitated (see the condition).
• The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.

So any method of inflicting the incapacitated condition on the mind flayer will end the grapple. Likewise, anything that gets the barbarian out of the mind flayer's reach will also end the grapple. There are far too many ways of achieving these to list here.

It's also worth mentioning that, since the Extract Brain ability requires the mind flayer to use an action, preventing the mind flayer from taking actions would also work. Most of the methods to do this would involve inflicting the incapacitated condition, and therefore end the grapple anyway. But as an alternative option, there's always mind control. There are a large number of spells that could be used to persuade the mind flayer not to extract the barbarian's brain, or just to let them go.

Since the mind flayer's grapple requires a Medium or smaller creature, polymorphing the barbarian into something Large should also free him. There is a minor point of ambiguity here - while the grapple can only be used on a Medium or smaller creature, it doesn't technically say anything about what happens if a grappled target becomes Large. But I think most DMs would agree that this should work.

The important part of that attack option is this:

And the water weird can't constrict another target

This same terminology is used of creatures like the Tarrasque who have the 'Bite and Swallow' feature or a Kuo-Toa Whip's Pincer Staff.

This means that the Weird cannot Constrict any target other than the one that it is presently constricting. So it can repeat this attack against the same target every round, in addition to the fact that you are being held underwater. Most likely, it will kill you by crushing rather than killing you by drowning.

In fact, the lore comments on it even makes it clear that it is trying to crush you.

[...] using its coils to crush any creature other than its summoner [...]

One other point of note when fighting a Water Weird: It has a 10' reach, and can fight just fine in deep water. The party may have a difficult time reaching it if it is sitting in water too deep to stand in. This can make it a little extra challenging as well if the party has to jump in and try to swim in order to fight it in melee.