Once I’ve unlocked BfA World Quests for this toon, what do I need to do to unlock it for alts?
That would include Mechagon, Nazjatar and Uldum and the Vale of Eternal Blossoms.
And what about Legion, including Broken Shore and Argus?
You will be shocked - Shocked, Ah Say! - to find that the unlock is entirely different and illogical for each of them. 
For BfA, the requirement seems to be covered in this comment here:
It agrees with what I remember.
For Legion, you first need to get your Flight Whistle from Uniting the Isles, and then I think Broken Shore is unlocked. I can’t remember whether there is an extra step for alts to get to Argus.
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For BfA, alts can skip unlocking the footholds on the other continent (although it’s still worth picking up one so you can actually go there) and the rep requirement for Uniting Kul Tiras/Zandalar. There aren’t any skips for unlocking the rest of the content.
For Legion, they can skip the rep for unlocking world quests, and there’s a dialog option to talk to Khadgar to skip the Broken Shore scenario to unlock the Broken Shore. There aren’t any skips for Argus, although your alts will get access to some stuff like the weekly quests and some of the world quests without the story requirement once you get there.
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