Did you finally figure out the solution? I am completely stock, not rooted and have the same issue since last 6 months. I never had any lock screen, so thats not the issue, I always have my phone unlocked.
The problem is not app specific. The delay in notification happens all across the board. When my phone is sleeping, i dont get any notification. As soon as I wake it up, all notification come at the same time, including Yahoo mail, Gmail, hangout messages, whatsapp, Google voice, twitter and others.
It seems like the wifi/data is disabled when phone is sleeping. I double checked in the Advanced Wifi option that wifi is always on. Some people said that the push notification heartbeat might be the issue but why only few of us are seeing this issue and not everyone?
Anyone have any solution?