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.
That being said, there are plugins available in the WordPress Plugins section that specialize in the ability to change usernames. These are not plugins developed by WishList Products but they are available to try out.
This is a link to a search for “change username” in the WordPress Plugins section: