I doubt the change to your PS4 name will affect your forum name here as I would not expect them to be connected. I could be wrong since IDK as the name implies.
While I could ink some information I will merely tag @ZOS_BillE since they probably not better if there are special cases for consoles.
Good luck to the person who will soon be formerly known as Devildog.
Changes to your PSN ID or ESO UserID will not affect your forum username. There is an ongoing thread linked below where you can post and request for your forum username to be updated.