FWIF, I am ALSO having problem with push notifications for certain 3rd party apps: Nest, Outlook, etc.. No badges. Tried all the usual routes: uninstall app, disable/enable notification, reboot, etc.
This happen with the same apps on 3 different devices: iPhone 6S+, iPad Pro 9.7 and iPad Pro 12.9. This seems to indicate that this is an issue with Apple's backend server, since it affect multiple 3rd party app on different devices but one common thing --> all affected apps rely on Apple's backend server.
Also, when I upgraded to 9.3.3, my iPad Pro now received delayed notification on built-in apps like Messages and Mails.