hangouts not syncing from PC to Nexus 6
It looks like GCM Push Notifications don't work with IPv6. I disabled IPv6 entirely in my router and now everything is fine. Though IPv6 was enabled for the past two years and I didn't have any kind of issue. Let's hope a Play Services update fixes it. 

Here is where I got the answer. On rooted devices, you can install [ROOT] Disable IPv6 to fix the problem.