I'm going to take a markedly-different approach from many of the others; hopefully this perspective is useful, too.
Start by looking at the nearest-neighbor spell to augury: aura of life. 30' radius, resistance to necrotic damage, regains 1hp at start of 0hp turn. Very easy, and the only thing really left up to interpretation is "nonhostile."
Most spells are like this: they're very clearly... spelled out, using defined terms. [/rimshot]
Augury is a different type of spell.
Augury is a super-vague spell, with lots of room for GM interpretation, preference, style, &c. And you're a first-time GM, so you feel like you've got little guidance from the spell. You're right, by the way.
There's a reason for that: while most spells are a manifestation of the player character's will, this spell is designed to do something fundamentally different. Augury allows the player character to engage in conversation with the GM. Not the player; the player character.
I want to be clear on this: the purpose of augury (and some other divinations) is markedly different than most other spells. Once you realize that, things fall into place.
At some tables it's perfectly appropriate for a GM to say to the players "hey guys, if you take on this dragon you're all probably going to die." If necessary, this is justified in-fiction as "your characters are noting the number of dried skeletons of previous adventurers, they know stories of Smaug from their childhood, they've trekked through miles of desolate wasteland just to get here." Or it's hand-waved away.
At other tables that would be completely unacceptable. It'd ruin immersion, it'd destroy people's roleplay, it'd cheapen the game. Crossing the boundary between in-world and at-table knowledge/conversation/interaction is forbidden.
Augury explicitly crosses that boundary, in a way supported in the fiction. Augury allows the character to talk (specify course of action taken very soon) and the GM to respond: "that should go well," "uh, I wouldn't do that," "kinda mixed bag," or "meh."
But what about timing, subject, and adjudicator?
My best advice is never to spend more than 30 seconds on an augury. Listen to the question/proposition, lean back, think for five seconds, and give an answer. If your players or PCs want better answers, they can wait (or pay) for divination, commune, or contact other plane.
You can use a reaction to affect initiative rolls.
Initiative (PHB 177):
At the beginning of every combat, you roll initiative by making a
Dexterity check.
You would automatically become part of the combat, but there is not a single instance, in the rules, that necessitates that you can't start the combat with a reaction with the usual exception of Surprise (PHB 189):
If you’re surprised, you can’t move or take an action on your first
turn of the combat, and you can’t take a reaction until that turn
ends.
Reactions (PHB 190):
Certain special abilities, spells, and situations allow you to take a
special action called a reaction. A reaction is an instant response to
a trigger of some kind, which can occur on your turn or on someone
else’s.
If you worded the trigger correctly, you would intervene in the combat and therefore would have to roll initiative.
Initiative (PHB 189):
When combat starts, every participant makes a Dexterity check to
determine their place in the initiative order.
You couldn't take another reaction before it is your turn.
When you take a reaction, you can’t take another one until the start
of your next turn. If the reaction interrupts another creature’s turn,
that creature can continue its turn right after the reaction.
Best Answer
You use your reaction before the roll
Both weal and woe rely on the same trigger (added emphasis mine):
The feature states you can use your reaction when the creature is about to make the roll, not when that creature is making the roll (which would require clarifications, like what happens with other features like the Bard's Bardic Inspiration) or when the roll has been made.
This reaction happens before the roll (its trigger).