Why does my iPhone not want to update to iOS 15.5?

Your phone needs to be connected to the Internet, either by Wi-Fi or cellular data, in order to update.

If any of the following are true, your phone may not be able to update:

• The specific phone model does not support the update.
• The phone has been "jailbroken."
• The phone is running an iOS beta. If so, then you will need to remove the beta profile first.
• The phone does not have enough free space. Typically it should have at least 5GBs available to update.
• The phone is configured for either or both Low Power Mode or Personal Hotspot.
• The phone is not connected to power and it has less than 50% charge left.
• The phone is not connected via Wi-Fi to the Internet. It won't update on a cellular-only connection … unless it is on certain 5G cellular networks.
• The phone is connected via a VPN.

I suggest starting with this support article: If your iPhone, iPad, or iPod touch won't update - Apple Support

Your phone's screen still shows that it has "No Service." Try temporarily connecting your phone to a Mac or PC to perform the update.