I had this same issue and found a fix for it.
In Settings>Notifications>Mail look for the options for Favorite mailbox. If this option exists than you have a mailbox set as a favorite. This option seems to default to off. When all other mail notifications are set to on and this is set to off, you will get a sound, and vibration, but no banner, no notification in Notification Center, and no notification on an Apple Watch. As soon as I turned this on, all notifications started working. I then deleted my inbox as a favorite, and this option went away completely and notifications continued to work.
To delete the favorite, open the app Mail and touch Mailboxes in the top left corner and then Edit in the top right. In the first section you will see a list of folders (Inbox, VIP, Flagged, etc.) that will have blue checks, this indicates what folders you will see in your mail app. At the bottom of the first section, click the Add Mailbox... option. You will see a list, if any of them are chosen, indicated by a grey check mark, they are selected as favorites and will only notify if you turn on Favorite notifications as mentioned above. Or you can simply uncheck them and standard notifications will apply.
Hopefully this helps some of you