Hi ChulaVJudy,
From the site Use your Apple Watch without your iPhone nearby:
"When your iPhone is off or out of range, your Apple Watch can use a Wi-Fi network to send and receive data. Your watch can also connect to a cellular network if it's a cellular model. And if you've set up an Apple Watch for a family member, they can use a cellular or Wi-Fi connection with their watch.
When your Apple Watch is connected to Wi-Fi, the Wi-Fi icon appears in Control Center. When it's connected to cellular, the cellular icon appears."
Are you able to confirm that the cellular icon is not displaying during the situations you describe? If your Apple Watch is not operating as described, it may be necessary to use Apple Support to work with AppleCare to investigate this further.
Regards.