iPhone not getting audio alerts from texts and other notification events
When your Apple Watch is connected to your phone notifications do not light up or make sound on your iPhone: this is unnecessary because you are getting the notification on your watch, so a phone vibrate/sound would be superfluous given that the notification is on your wrist. Thus, the issue is not with your iPhone (as it is not supposed to vibrate when it gets a notification with Watch connected). 

So, why is your Apple Watch not vibrating?

Make sure you have not turned off haptics.

Make sure you are not in Silent Mode on your watch.

Turn up the intensity of the haptic vibration alerts on your Apple Watch to ensure that you can feel it when it vibrates.

If you're still having an issue, try restarting your phone and watch  or unpair and repair your watch from your phone (this has helped many users solve this problem).

It's also possible that your Apple Watch's haptic (vibration) engine is broken or weak. Apple acknowledges that this can happen sometimes. Head to apple support at getsupport.apple.com, go to Apple Watch, and press the issue section "Repairs and Physical Damage". Then go to the issue "Taptic Engine or haptic feedback is not working as expected," where you can mail in your Watch for repair or talk to Support on the phone about misfunctioning vibrations.