Take the portal to Dalaran and ask the innkeeper and it will pop in your bag like the normal one.
I was doing Shadowlands timeline and wanted the Dalaran Hearthstone so I did the following:
Went to Org
Talked to Chromie and selected another timeline - Choose “Fight the legion”, accept the quest that automatically pops up. Go to entrance of Org and turn in this quest. Choose skip intro. Auto ports you to Dalaran where right in front of your spawn point you should see a person to talk to who gives you the hearthstone immediately
Bumping this after 10.1.5 - the hearthstone is now a toy - you don’t have to do anything to get it aside from open your toy box 
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This is true, if you already have one on another character.
If you do not have one on another character, and you do not want to, or can not, take the Legion start quest, you can go to the Broken Isles portal that takes you to Aszuna. From there you fly to Dalaran (it is nearby), go to the Inn in the center of the city, and talk to one of the NPC’s. I believe it is the assistant innkeeper, but I might be wrong on that.
If you can, and choose to do so, you can also take the Legion start quest (Note: You can skip the whole opening battle scenario by selecting that option) in the conversation with Recruiter Lee. After either skipping or running the scenario you end up in Dalaran, and in front of you is a mage with a quest. That quest grants you the Dalaran Hearthstone.
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Sure, I guess I should have mentioned that. Thanks for pointing it out.
ETA: Go to Legerdemain Lounge (the faction-free inn in the central buildings) and talk to Amisi Azuregaze, who is near the stairs to go up. She should have a dialogue.
She doesnt offer a hearthstone.
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IIRC, the dialogue to set your hearthstone to that location gives you the dalaran hearthstone.
I think they changed it now where the Dalaran hearthstone becomes a toy so each character has one as long as you’ve gotten it before on someone else.
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Yep. The Dalaran and garrison hearthstones are now toys. I’m curious where my evoker will go if he uses the garrison hearthstone toy.
if you haven’t unlocked a garrison, you can’t use the garrison hearthstone.
Thank you for this!