In order to start the 8.2 storyline into Nazjatar, you must have world quests unlocked. Once you get those unlocked, the quest you see popping up in the youtube videos will pop up for you as well. (You may have to relog in order for it to activate if it doesn’t immediately.) Information on how to unlock world quests can be found here.
You need the quest “Uniting Kul Tiras” (or “Uniting Zandalar” for Horde.) You should be able to pick it up from Halford Wyrmbane on the ship with the mission table.
You also need to complete the 3 foothold quests for the other continent. (So, basically follow the War Campaign story until you have access to all 3 zones of the enemy faction.) These quests also start from the ship with the Mission Table, and you may have done some or all of them while leveling up.
What others have said is correct, but the simplified version is…
do each of the War Campaign opening scenarios (1st quest chain in each of the opposing faction areas) up until the point they send you back to Boralas.
after this you will get a quest on the boat to get each alliance bace rep to friendly (you don’t need to complete the full zone - about 1/3 of each on the main quest line will get you there) Proudmore/Storm’s Wake/Embers - looks like you only need Embers (Drustvar)
This will unlock WQs
Then you should get the quest to go to Nazj - you may need to log out and back in. Its on the boat, but I think you can get it as a pop up or from the mission board/dungeon journal