Here's my experience, just to keep things confusing for everyone:
After upgrading my Series 1 Watch to watchOS 5 and my iPhone 8 to iOS 12, I had this problem: Arriving emails would make my Watch vibrate, as expected, and if I immediately raised my wrist, the arriving email would open on the Watch screen. But if I didn't look at the Watch immediately, the notifications didn't stick. In other words, I'd feel my Watch vibrate and I'd make a mental note to check it later. But a few minutes later, when I'd look at the Watch, there would be no indication of a notification — no red dot, and nothing listed when I swiped down to check the Notification screen. This only happened with email. All other notifications appeared to be working as normal on the Watch.
After erasing my iPhone and setting it up from Recovery Mode as a new phone and setting up the Watch as new: Notifications for emails sent to my iCloud account work perfectly on the Watch. Notifications for emails sent to my other email account (Fastmail) do not work at all, not even the initial vibration. Notification settings for both accounts are identical and are set in the Watch app to mirror the phone.
It should be noted that I did the restoring of the iPhone and the Watch very soon after the watchOS 5.1.1 update. So it's quite possible that the 5.1.1 update made the difference, but I didn't have the presence of mind to check before erasing both devices and starting over, because I had a few other reasons to take that drastic measure, not just this notifications issue.