Hi all, just joined regarding this issue. iOS 10.0.2, iPhone 6.
Since updating last week, iCloud email has NOT been pushed to my phone consistently, and specifically when on cellular network. When connected to a Wi-Fi hotspot, push seems to work.
On a related note, when push for iCloud *is working*, the email alert sounds for my iCloud email work. When push is not working, and I have to open the email app to *fetch* new emails, there are no alert sounds when the emails do show up (in real time on the app).