Question

How long must Shuckle hold Berry Juice before it turns into Rare Candy?

Answer 1

There is not a single mention of Berry Juice being able to turn into Rare Candy in the Gold and Silver source code, so it's pretty safe to say that this was nothing more than an urban legend in the first place.

Pokegold (GitHub)

Answer 2

It doesn't happen in any generation apart from Gen II, and there's no definite time frame. It just says "over time", the Berry Juice will harden into a Rare Candy if held by a Shuckle.

Comments

There has to be some formula or set value or range of values for this. Points to whoever can figure it out!

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Actually, someone already did: http://asia.gamespot.com/pokemon-black-version/forum/c-d-shuckle-should-have-its-berry-to-berry-juice-to-rare-candy-ability-returned-61941787/

Different berries take different amounts of time, apparently. I didn't link it because it's a forum post (reliability), and not a site post like Bulba or Serebii.

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Last time I checked, Sitrus Berry wasn't in Gen II. :P
That's a theoretical "What Gamefreak could do with reintroducing this with the new berries" not any actual numbers.

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I need to read things more carefully. :P