Yup, same here.
I've put all my eggs in Apple's basket, using iCloud only for email, and my old Google account automatically forwards to my iCloud now.
I too have the push issues. Sometimes they come through no problem, other times I'll open up the mail app on the iPhone 7 Plus, and an email from the night before will refresh. It's a real problem, as I'm normally very punctual with my responses.
I have however noticed that if I'm on my MacBook, the email notification will popup on my computer, and not my phone to avoid multiple devices sounding off. However this isn't always the case, as it happens when I'm out of the house as well.
I really hope someone finds a solution, or Apple pushes a fix. Has anyone reported this as a bug to the iOS bug report submission forum?