Have you seen Product Update - What you need to know?
Note: If Nintex Workflow for Project Server 2013 is installed in the same environment, restart the "Microsoft Project Server Queue Service 2013" service and do a final IISreset on each server.
In my case the service unavailable HTTP 503 error showed up on all sites except for the search, which is obviously on a separate web application. Which made me think maybe this was related to web application/application pool level since IIS reset did nothing. I opened the IIS Manager and checked the application pool to find most of it to be stopped. Everything started to work after i started the application pools manually.