Voice of Authority is used immediately after (a) the spell is cast and (b) you select one of the allies (perhaps the only ally) you have targeted in casting the spell.
You offer four possible alternative timings:
- Is cast.
- Targets the ally.
- Affects the ally.
- Ends.
There is no difference in the timing between 1 and 2
If a spell requires a target, choosing a target is part of the casting of a spell. The rules say "A typical spell requires you to pick one or more targets to be affected by the spell's magic." (PHB 204, emphasis added) A DM might, for laughs, allow a player to cast magic missile to "attack the darkness", but if a spell requires you to pick a target, then if you don't select a target, the spell doesn't get cast. I can offer examples to back up this claim, but establishing this further would be a separate question.
If the spell only has one target, case closed. If there is more than one target, you get to select which one is affected by Voice of Authority. But "choosing a target" and "casting a spell" still happen at the same time.
3 is not the correct timing, because you don't target a creature every time your spell affects a creature
You raise the issue of an ally being affected by the same spell multiple times. At one point in your question you use the construction "targeted/affected" as if they are interchangeable. But there is a difference between a creature being targeted and a creature being affected.
I think that the area of effect of dawn introduces complications (which I will address below), so I'm going to switch from your example of dawn to heroism, a spell on the Order Domain spell list. Unlike dawn, which has an area of effect, heroism targets a creature you touch and the spell clearly refers to the creature as "the target". The creature "gains temporary hit points equal to your spellcasting ability modifier at the start of each of its turns." This creature is the target of the spell, and is affected multiple times. Does Voice of Authority trigger each time they are affected?
No, because the act of choosing a target only happens once, when the spell is cast. Voice of Authority happens "If you cast a spell with a spell slot of 1st level or higher and target an ally with the spell. [emphasis added]" Each turn when the creature gains temporary hit points, they are affected by the spell, but you have not targeted them an additional time. You targeted them once, when you cast the spell. There is nothing in RAW that suggests that you have somehow targeted them an additional time, or that targeting and spellcasting are separated in time.
But you also raise the issue of being the target of an area affect spell (specifically, dawn). You write:
For simplicity, I'll assume that all creatures within the area count as targets.
You link to a proposed answer that has not been accepted as the correct answer. While you can make that assumption, it has not been established that the proposed answer has correctly identified RAW. I think the proposed answer is incorrect, but I don't think it has to be answered in order to answer the current question. The discussion regarding heroism establishes that the spellcaster targets a creature when the spell is cast, not when the target is affected.
4 is not the correct timing, as it would rely upon an extremely unnatural reading of the phrase "immediately after the spell".
Finally, you have the difference between 1 (when the spell is cast) and 4 (when the spell ends). To see the implications of this, consider the spell aid. This can be used to target an ally when the spell is cast, and has a duration of 8 hours. As it does not affect the ally at multiple discrete times, it can only be interpreted as when the spell is cast or when the spell ends.
It would be extremely strange if the intention of Voice of Authority would be for the ally to use their reaction to make a weapon attack 8 hours after casting the spell, when you will most likely not be in combat.
Compare the actual rules for the Voice of Authority feature with an alternative wording (difference in italics):
The actual rules: If you cast a spell with a spell slot of 1st level or higher and target an ally with the spell, that ally can use their reaction immediately after the spell to make one weapon attack against a creature of your choice that you can see.
Hypothetical alternative wording: If you cast a spell with a spell slot of 1st level or higher and target an ally with the spell, that ally can use their reaction immediately after the spell's duration ends to make one weapon attack against a creature of your choice that you can see.
The most natural reading of "immediately after the spell" is "immediately after the spell is cast". "After" is somewhat vague. "Immediately after" is a little clearer. Still, it's not impossible to read "immediately after X" as "immediately after X ends". But given an alternative interpretation, like "immediately after the spell ends", the reader should ask, why doesn't it say that? I suggest that your proposed reading is unnatural, and an English writer/editor would realize this and write something clearer, like the hypothetical alternative wording I have supplied. Since spells can have durations measured in minutes, hours, days, or longer, Voice of Authority would be useless for most or all non-instantaneous duration spells if we adopted your proposed reading.