SOLUTION (confirmed working for horde-to-alliance-xfer in 8.3)
For Boralus: Go to Stormwind, pick up Tides of War. Follow Jaina to Boralus, skip all the scenarios if you want. Follow the intro quests in Boralus until you complete the quest “Sanctum of the Sages”. Wait for the Stormwind portal to appear - go through it, load into the Stormwind portal room, turn around and take the Boralus portal back (it should be working now.)
Now you’ll probably discover that you can’t get into the Chamber of Heart either. If this is the case, go to the campaign ship in Boralus harbor, talk to Greymane and start the quests to go to Nazjatar. Follow in the Nazjatar introduction until Jaina helps you open the portal back to Boralus. Magni should now give you a quest to meet him in Silithus to talk about Essences. Go through the Naz portal to Boralus, then to Silithus, and now your Heart Chamber door should be working. Get the quest to go to Highmountain, do that whole thing, come back, and you’ll get the 8.3 campaign quest from Anduin as soon as you finish equipping your first Essence onto your Heart.