If alerts for emails and/or messages are not being delivered on your Apple Watch as expected and as detailed here:
Notifications on your Apple Watch - Apple Support
Then the following steps may help:
Check for and install any available updates for your iPhone and Apple Watch, starting with your iPhone:
Check that Wrist Detection is turned on and that, if passcode is enabled, you are unlocking your Apple Watch after placing it on your wrist
- On your iPhone, in the Watch app, go to: My Watch (tab) > Passcode > turn Wrist Detection on.
Check that you are wearing your Apple Watch sufficiently snugly that it does not believe that it has been removed (which will cause it to lock automatically):
Check your Mail settings:
- On your iPhone, in the Watch app, go to: My Watch (tab) > Mail.
- Select Custom.
- Turn on Show Alerts.
- Tap on your email account name (eg iCloud) > turn on Show Alerts from iCloud > turn on Sound and/or Haptic, as desired.
- Return to the main Mail options page and tap on VIPs to configure your alert options for any contacts that you may have designated as VIPs (within the Mail app on your iPhone).
- Return to the main Mail options page and tap on Include Mail > select the mail folders to be shown in the Mail app on your Apple Watch.
Check your Messages settings:
- On your iPhone, in the Watch app, go to: My Watch > Messages.
- Choose Custom and check - under Alerts - that Show Alerts, Sound and/or Haptic (as desired) are each enabled.
Check that Theatre Mode is not enabled on your Apple Watch:
Check that Do Not Disturb mode is not enabled on your Apple Watch:
- When viewing the time on your watch, swipe up to open Control Centre.
- Tap the Do Not Disturb button (crescent moon icon) to toggle the setting on / off.
On your iPhone, in the Watch app, go to: My Watch > Sounds & Haptics:
- Check that the Alert Volume is towards the centre or right hand side.
- Check that Silent Mode is not enabled.
- Check that Haptic Strength is set to the middle or far right hand side of the slider.
Confirm directly on your Apple Watch that Silent Mode is not enabled:
- When viewing the time on your watch: swipe up on the screen to open Control Centre > check whether Silent Mode (button with a crossed out bell icon) is enabled.
Check that your Apple Watch and iPhone are connected:
The following further steps may also help:
Restart both your iPhone and your Apple Watch, turning both devices off together, then restarting your iPhone first;
Unpair and re-pair your Apple Watch:
If the issue persists, then I suggest that you contact Apple Support again (mail-in service may be available, if required), make a Genius Bar reservation or visit an Apple Authorised Service Provider for further assistance, if necessary including to arrange hardware service:
Apple Watch Service Answer Center - Apple Support