PokéBase - Pokémon Q&A
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My opinion is that we already have way too many questions about this topic, some of which have answers that work for 1 or 2 generations, so I want to get the answer for every variable and every generation in one place. For eggs, are they determined when the day care person finds the egg, when the player receives it, or determined like this? For roamers, are they determined when it starts roaming, when the player encounters it for the first time, or at the start of each encounter? Can the player re-roll the variables by defeating the roamer and then defeating the champion? Are different variables determined at different times, ie. is it true that in some generations, personality is determined when the day care person finds the egg, and IVs are determined when the player receives it?

Stuff I think I already know:
when natures and personality are determined for eggs in every generation source source
when IVs are determined for roamers in Crystal source
when IVs and natures are determined for roamers in Generation 4 source
how swap breeding works in Generation 7 source

Please don't question my curiosity.

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I don't know for all the regions, but for Gen 8 Sword and Shield, the egg values (including whether or not it is shiny) are determined when the nursery worker folds her arms to indicate she has an egg. For the roamers, (the Galarian Birds) it is determined when you encounter them, so you can keep resetting and catching till you get the best IVs/EVs/Nature that you want. Hope I helped! :D
NOTE: This isn't an answer because you asked for all the generations, and this is just Sword and Shield, but still :)

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