Ok, I found the issue. 
To handle htaccess and IP filtering within the admin UI of the hosting system, we placed Varnish (without any rules) before Nginx. However, since Varnish is not able to handle websockets, it fails (even with the workarounds mentioned in the Varnish documentation).
I turned off Varnish and moved the IP filter and htaccess configurations into Nginx.
Now everything is working!