Bringing Order to the Isles

So basically if you are curious here's what I did to get a point for the achievement

1- Finish your class order hall Campaign ( up to "A Hero's Weapon" )

2- Skip to the Broken Shore and pick "Champions of Legionfall" and start the questline until you get your new follower ( no need to do quests for Breaching the Tomb just pick what I mentioned )

3- After getting your new follower go back to the Broken Shore and pick the class mount questline in broken shore

Script to check your progress WITH CLASSES:
/run local t={108360,108376,108651,108655,108654,108648,108656,108652,108657,108653,108650,108649}for i,id in pairs(t) do local _,_,c = GetAchievementCriteriaInfoByID(42565,id)print(GetClassInfo(i),c and"\124T136814:0\124t"or"\124T136813:0\124t")end

Thank you for this beautiful journey everyone it was fun updating this comment and communicate with you all see you soon <3

For anyone who wants this info all tidy like so you don’t have to go searching for it…

You only need to complete Breaching the Tomb on one character. That’s the achievement for doing the entire Broken shore questline. Don’t do it on every alt.

You will need to complete your main class hall questline (until you get your class title basically, plus an extra quest or two in your hall, the one where you have to stand in the spot and hold your weapon up) (unlock WQs from Khadgar too just to be safe).
After that, Khadgar is waiting on Krasus Landing with the Legionfall intro quest. You need to complete it (skip the scenario) and turn it in at Deliverance Point.
After that, complete Champions of Legionfall from Maiev at Deliverance Point. This one should autocomplete and you can turn it in immediately to her.
Then, you should be able to go back to your class hall and there will a new shield quest available from one of your class leaders (Mograine for DKs, Ravenholdt for Rogues, etc.).
Complete that quest line until you recruit your new champion (Minerva Ravensorrow for DKs, Lillian Voss/Tess Greymane for Rogues, etc).
Go back to Deliverance Point on Broken Shore and there will be a new shield quest available from someone class related who isn’t part of the Khadgar, Maiev, Illidan group (Dread Commander Thalanor for DKs, Apothecary Keever for Rogues, etc)
This will lead you back to your class hall for the questline that gives you the class mount.
Some classes have other color variations that can be obtained in other ways, like Warlock has three, the second you get from a separate achievement/purchase, the third is a drop from a rare, I don’t know if they changed this for Remix.
Things you DON’T need to do:

Balance of Power questline
Illidan questline
Light’s Heart questline
Breaching the Tomb on each character
Sacrifice your first born child (seriously don’t do this)
Each zones questline
Anything in Suramar
Collect more than one of your artefact weapons
ADDENDUM: It’s been brought to my attention that some of the actual follower/mount questlines do require some other quests to be completed, such as Paladin requiring some quests in Suramar until you can get 500 ancient mana. So your mileage may vary and sorry for potentially leading you on based on your class.

Modified version of Witnesscm's script that also shows class names (at the time of writing the script in the quoted comment only giving you "criteria #" instead of a class name):
/run local t={-288,-272,3,7,6,0,8,4,9,5,2,1}for i,id in pairs(t) do local _,_,c = GetAchievementCriteriaInfoByID(42565,108648+id)print((GetClassInfo(i)),c and"\124T136814:0\124t"or"\124T136813:0\124t")end
EDIT 20:45 UTC: The FoS is now obtainable (at least on EU). If you previously completed shaman/monk class order mount questline, you need to log in on those characters to register achievement criteria.