Explaining it as simply as possible, talents happen when they say they happen.
Keeping in mind that the Rules Reference specifies in the Attack section (pg. 2) the following: If a talent is triggered “when attacking,” its effect occurs simultaneously with the damage being dealt, and before retaliation. If a talent triggers “after attacking,” it occurs after resolving the attack and retaliation if it is not defeated.
If a talent says it happens as a result of moving onto a specific hex, the talent triggers right after the unit moves onto that hex, it doesn't wait until the full movement is over.
If something triggers off of being reduced to 1 HP, it triggers when the unit reaches 1 HP (keeping in mind that as above, attack/retaliation happen back-to-back unless the talent says it happens 'when attacking' or that the retaliation is skipped).
For a specific example, let’s look at an AI Taproot as I know that’s been discussed. Let’s say it has 2 HP and 2 movement. It moves 1 hex and triggers Toxic Secretion, losing 1 HP. It now has 1 HP and so the Taproot’s Summon is triggered. It doesn’t finish its movement, it uses Summon, and gets flipped and grouped under whatever it summoned. The new unit also doesn’t finish the Taproot’s movement or move itself, because the AI Summon talent says it doesn’t move this turn.
Now, if the Taproot were to be reduced to 1 HP from an attack, the Taproot would get to retaliate first (because that’s part of the attack and Summon says ‘after’), and then it would Summon.
So ya, overall I would think about it less as interrupting and more just as, things happen when they say they happen.
With regards to “immediately,” its use doesn’t usually have a special meaning, it’s usually just used to emphasize unusual timing, but timing that would occur even if that word were not in the sentence.
For example, Entropy: “If this chip’s health is reduced to exactly 1, it does not retaliate and defeats itself immediately, dealing 2 damage to all adjacent non–naroran units.” If you take out the word immediately, this talent works exactly the same, immediately just feels right in this sentence to emphasize that something is happening with somewhat unusual timing than normal.