Ok after doing some web searches. In a developer article with web sockets I found a recommendation to also install the HTTP Activation feature when using WebSockets. Using powershell we did the following.
Install-WindowsFeature -name NET-HTTP-Activation
Install-WindowsFeature -name NET-WCF-HTTP-Activation45
Do this seems to have allowed WebSockets to start working. I am not sure what the two above installations install in addition, but something there fixed the WebSockets issue for now. Maybe these are required and are there by default on 2012 r2 +...
Maybe the installer should be updated to check for them. Installing an update and breaking a major function like mail refreshes hurts.
We are doing more tests to ensure the issue is fixed for all customers, but right now it looks like we are at least closer to the root issue.
If further testing holds I will come back and mark this as the Resolution.