The rules as written are ambiguous - and so it's up to the DM's ruling (and designer clarification has gone both ways)
Unfortunately, it's not clear exactly how these rules interact. The troll has an ability which apparently specifies the only circumstances in which a troll dies, and these spells can cause creatures to die instantly; both are exceptions to the normal rules about how things can die, so the principle that "specific beats general" guides us in trying to resolve the conflict, but judging which of the two features is more specific seems subjective and ambiguous.
If you interpret the spell's rules as being more specific than troll regeneration, the troll dies; and since it is dead, regeneration becomes meaningless. If you interpret the troll's regeneration as being more specific than the spell, it precludes the death from happening despite the spell's effect.
Purely RAW ruling, on specific-beats-general principles, I would personally read the troll's ability as more specific than the spell; the spells can, after all, be used on many different kinds of creatures by many different casters, but a troll's regeneration is only ever about trolls, so it is necessarily much more limited in scope and therefore takes precedence over the rules of the spells.
JC says the troll dies
Official D&D 5e rules guru Jeremy Crawford weighed in when Rubiksmoose asked him about this issue on twitter:
Rubiksmoose: Trolls say that they die "only if it starts its turn with 0 hit points and doesn't regenerate". Does that mean they cannot be killed by power word kill/divine word? How about Disintegration?
Jeremy Crawford: If you're affected by the power word kill spell, it doesn't reduce your hit points to 0. It kills you. It thereby bypasses features that rely on you having 0 hit points. The disintegrate spell does reduce hit points, but if it reduces you to 0, you're dusted.
In this case he's just repeated previously given clarification that effects which state creatures are killed or die do not function by reducing the target's HP to 0, they just kill the target directly. This fails to address the actual cause of the ambiguity, so we can't really take it as a clarification of the rules as written.
The response in context implies that Crawford believed that this aspect of the troll's regeneration ability is only meant to stop the troll dying due to normal hit point damage (as a "feature that relies on you having 0 hit points"), not prevent any other effect which reasonably causes death.
And then JC says the troll doesn't die
When asked a very similar question on a later podcast, about whether or not the instant death (massive damage) rule could kill a troll, he suggested that it should not:
Jeremy Crawford: So if we're gonna use the troll as the example, here's what we're told: the troll regains 10 hit points at the start of its turn; if the troll takes acid or fire damage this trait doesn't function at the start of the troll's next turn; the troll dies only if it starts its turn with zero hit points and doesn't regenerate.
So in D&D the specific beats the general, and the massive damage rule is a general rule, and here we have the specific troll. But let's also look at the massive damage rule, because sometimes a general rule in the way we might think it does if we're just going off our memory of it...
[... looks up rules, reminisces about killing player characters ...]
So looking again at the instant death rule, the troll's exception overrides the general rule.
Bart Carroll: So it'll be smushed, but it will reform...
JC: It'll regenerate, yep, and that is part of the horror of the troll.
Instant death by massive damage doesn't kill you by dropping your hit points to zero; it just says you die. Dropping your HP to zero is a necessary precondition for this rule to apply, but the statement that "you die" isn't any more qualified than the effect of Power Word Kill. Coming back to it later, JC seems to have taken a more literal reading of the troll's regeneration ability and ruled that it really does only die if it meets the requirements specified by its regeneration ability.
It seems plausible that if he'd been asked specifically about trolls and Power Word Kill again, he might have ruled differently, depending on whether he thought the spell was more specific than the troll's trait. He does preface the judgement by explaining how loathe he is to make rulings about general hypotheticals, because D&D isn't a coherent rules system, and would probably argue if challenged that this ruling was specifically about instant death and trolls, whereas the previous was about Power Word Kill/Disintegrate and trolls, so they don't conflict at all.
Best Answer
Juiblex lives
One of the core guiding principles in 5e is that the specific overrules the general:
To understand how to resolve this, it is important to understand the basic rules so you can see which parts are being changes.
Normally when you are reduced to 0 hp, you are knocked unconscious:
How do Juiblex and the Nightwalker modify these rules?
We can see that Juiblex is an exception to this rule, it "dies only if it starts its turn with 0 hit points and doesn't regenerate." This means that Juiblex will survive any number of failed death saves, so long as they can regenerate.
The Nightwalker also overrides some of these rules. Usually when a creature is reduced to 0 hp it is unconscious, unless it suffers instant death, and then it has to fail 3 death saves to die. However when reduced to "0 hit points from damage dealt by the nightwalker dies and can't be revived by any means short of a wish spell."
So what happens if Juiblex is reduced to 0 hp by a Nightwalker?
Well, Juiblex modifies the instant death and saving throws part of the rules. But the Nightwalker modifies what happens when you are reduced to 0 hit points.
The key point is that the Nightwalker's ability makes something die. Normally something that dies, becomes dead. However, Juiblex has a special ability that modifies this expectation. In this case, Juiblex's ability modifies the general rule, so Juiblex survives.