by keeping your visitors in mind, in the normal case, the changing password process should not require an outage , meanwhile, at Production environment , you should apply it out working hour.
My suggestion is to apply this process first at Test, but in your case, the services account is the same in both environments , so you should care of that.
also, Try to list all application services , application pool , windows service and SQL services and make sure it's working and up running after changing the password .
Here you can find all possible issues that may face you and how to overcome it Managing Passwords for Service Accounts in SharePoint & SQL Server
The following are the required steps in normal scenario :
Navigate to SharePoint 2013 Central Administration interface, click Security in the left-hand menu, and click ‘Configure Managed Accounts’. Select the farm administrators to account in the account list shown, click the Edit icon and change the password.
Manually change the User Profile Service password. As required by SharePoint, this service uses the farm administrator account, however, SharePoint 2013 does not treat this account as a managed account so it must be changed manually.
Check if any applications in the Secure Store service use the farm administrator account, and if they do change the password there.
Reboot all the servers in the SharePoint farm, except for SQL server. SQL Server does not need to be restarted.
For more details check Step by Step: Changing the SharePoint 2013 Farm Account Password