Ansible: How to change MySQL server root password by reprovisioning the server
The problem you have is that Ansible is trying to use the same root password to login as you want to change it to:

- name: Set root user password
  mysql_user: name=root
              host="{{ item }}"
              password="{{ mysql_root_password }}"
              check_implicit_admin=yes
              login_user="{{ mysql_user }}"
              login_password="{{ mysql_root_password }}"
              state=present


Obviously this is never going to work if you want to use this play to change it.

Instead you should change the above play to be something like:

- name: Set root user password
  mysql_user: name=root
              host="{{ item }}"
              password="{{ mysql_root_password }}"
              check_implicit_admin=yes
              login_user="{{ mysql_user }}"
              login_password="{{ mysql_old_root_password }}"
              state=present


And then update the relevant inventory files to add this new variable.

So your group_vars/production should now contain:

mysql_old_root_password: productionpw
mysql_root_password: newproductionpw


It looks like this playbook uses the root password in both the roles/mariadb/tasks/main.yml playbook and also roles/wordpress-setup/tasks/database.yml so you might want to run the whole server.yml playbook to make sure this is set up properly.