WishList Member uses the WordPress user database to create Member accounts. This means the WordPress Lost Password, login, etc. functionality works for members.
This also means the Username for a member cannot be changed once it is set. This is because WordPress does not allow for usernames to be changed/edited after they are created.
That being said, there are plugins available in the WordPress Plugins directory that specialize in the ability to change usernames. We have not developed these plugins, but you can check out what is available as they can typically handle that task.
A search for “change username” in the WordPress Plugins directory provides some useful options.
