I have the exact same issue:
iPhone X iOS 12
Apple Watch 3 watchOS 5
As soon as I updated my phone and watch my mail notifications stopped working properly on my watch. Originally they were set to custom in the watch app with sound and haptic set to on. Switching to mirror my phone resulted in one exchange account getting sound and haptic but my second exchange account only getting sound. Both accounts were set to have the exact same notification settings and sound and haptics were set to the same tone and rapid vibration setting on my phone.
Ultimately I switched to the Outlook app for now for both accounts and they work as expected on my watch with both sound and haptic working again. Honestly notifications come in a lot faster too.
Some things I have seen said to change the vibration type to alert for the new mail section of sounds and haptics on the phone settings.
I also am an IT professional and support a fair amount of Apple devices. I rebooted both phone and watch multiple times as well as unpairing and re-pairing the watch, reset network settings, all with no change.
Feels more like a bug in the mail app/watchOS interaction.