The true polymorph spell, for your use of it, says:
Choose one creature...that you can see within range. You transform...the creature into a nonmagical object...
By choosing the creature, you have made the creature into the target of the spell. The object it transforms into is also under the effect of the spell. The original creature is still the target, even though it exists in a conceptual, not a physical sense. You know that the game treats the creature as if it still exists because it will be restored when you use dispel magic, and because the creature itself can be seen with truesight.
If you turn a creature into an object, it transforms along with whatever it is wearing and carrying into that form, as long as the object’s size is no larger than the creature’s size. The creature’s statistics become those of the object...
Since the creature's statistics are now those of the object, the creature's hp are those of the object as well. This is important because if the object's hp become zero, then the creature's hp become zero, and this ends the spell.
The spell lasts for the duration [Concentration; up to one hour], or until the target drops to 0 hit points or dies. If you concentrate on this spell for the full duration, the spell lasts until it is dispelled.
This spell has no effect on a shapechanger or a creature with 0 hit points.
There is some disagreement about whether concentrating on the spell for the entire duration means that the "or until the target drops to 0 hit points or dies" is still relevant. Fortunately a ruling there is largely irrelevant because of the next sentence; the spell has no effect on a creature with 0 hp. The target of the spell is the original creature. If the target of the spell drops to 0hp (which it will when the object is at 0hp because it has the statistics of the object), the spell ends. This is true even if the spell has been concentrated on for the full duration; it is an overarching condition that can end the spell1.
Normally the game does not worry about the hp of an object unless something is damaging them. The DMG section on statistics for objects (246, 247) focusses entirely on how to break or destroy them, including saying:
Hit Points. An object's hit points measure how much damage it can take before losing its structural integrity.
Damaging the object should at some point reduce the target creature to 0hp and thus end the spell effect. A DM could rule that snapping the twig means that it has "lost its structural integrity", reached 0hp, and ended the spell. Likewise, if conceptually it is no longer "the twig" then "the object" has been destroyed, and the spell would end. On the other hand, the DM could also rule that merely snapping the twig has damaged the twig without destroying it - 'the twig' still exists, it is now just in two pieces.
In the latter case, we have an identity problem, also known as a Ship of Theseus problem. Which of the two parts of the twig is now the object tied to the target creature? Or are they both? Individual DM's will have to work this out for themselves, according to the philosophical framework they want to exist in their games.
What is clear, however, is that with continued snapping, at some point the pieces will no longer be "the twig", or enough damage will have been done to reduce the twig to 0hp. At the point the object can be said to no longer exist2 or to be at 0hp, then the spell ends and the creature is restored.
If the creature had the statistics of the object, and was restored when the object reached 0hp, does that mean that they are at 0hp when they are restored? I don't think so, based on a comparative reading of the 'creature into creature' and 'creature into object' sections of the spell (emphases mine except for headings).
Creature into Creature...
The target assumes the hit points of its new form, and when it reverts to its normal form, the creature returns to the number of hit points it had before it transformed. If it reverts as a result of dropping to 0 hit points, any excess damage carries over to its normal form. As long as the excess damage doesn’t reduce the creature’s normal form to 0 hit points, it isn’t knocked unconscious...
Creature into Object. If you turn a creature into an object, it transforms along with whatever it is wearing and carrying into that form, as long as the object’s size is no larger than the creature’s size. The creature’s statistics become those of the object, and the creature has no memory of time spent in this form, after the spell ends and it returns to its normal form.
Note that the 'creature into object' description does not specify what hp the creature is at when it is restored, but only says it 'returns to its normal form'. Since the 'creature into creature' section occurs first, I believe that it is defining what 'returns to its normal form' means: 'the creature returns to the number of hit points it had before it transformed'. The 'creature into creature' section then further elaborates that the original creature takes the excess damage that went beyond what was needed to reduce the new creature to 0hp. The 'creature into object' section does not specify such a thing, so the excess damage rule appears to apply only to creatures.
Thus I would rule that when the twig is snapped, the object is at 0hp; this makes the original creature sharing its statistics at 0hp, this triggers the end of the spell and restores the original creature, which returns at the number of hp it had at the time the spell was cast and having not taken excess damage from the destruction of the object.
Further evidence that the destruction of the object is not meant to kill the original creature is the phrase, "the creature has no memory of time spent in this form", which would be unlikely to be necessary if it was dead.
1One might argue that "this spell has no effect on a shapechanger or a creature with 0 hit points" refers to targeting rather than duration, but this claim is not supported by context. First, if the clause was about targeting it should go earlier in the spell description, before the effects of the spell are described ("you transform") and not after the section on duration. Second, it does not say "This spell cannot target a shapechanger or a creature with 0 hit points". It does not say "You cannot choose..." It does not say "This spell will not affect..." (future tense). Rather, it says, present tense, this spell has no effect. Thus anything that reduces the original target creature to 0hp will end the spell effect.
Although I was unaware of it at the time I originally wrote this answer, this interpretation is supported by a JC tweet:
@DMJazzyHands once a true polymorph spell is made permanent from concentrating for the full duration, does reducing that creature to 0hp still cause them to revert to their original form?
@JeremyECrawford The text of the spell says it has no effect on a creature with 0 hit points. That statement is made after the bit about lasting until dispelled. At 0 hit points? The transformation ends.
2What happens to a spell effect when the object on which the effect is fixed is disintegrated, for example? (and cf. What happens to a destroyed Pact Weapon?) One could rule that the target creature was then no longer recoverable - if there is nothing left upon which one could dispel magic, they simply cannot be brought back. However, I think this goes against the spirit of the polymorph spells, which are not meant to be 'cheap' ways of disposing of enemies. I would rule that if the object was well and truly destroyed, that would end the spell effect and thus return the original creature.