iPhone 11, iOS 15
You can update your iPhone to iOS 16.4.1
See -> iPhone models compatible with iOS 16.
iPhone models compatible with iOS 16 – Apple Support (UK)
You need about 6 GB device storage and good Service and/or Wi-Fi.
If they are On, turn Off Hotspot & VPN.
Charge the battery > 70%.
Go to Settings -> General -> Software Update -> Download and Install.
See the following Apple articles:
Update your iPhone, iPad, or iPod touch
Update your iPhone, iPad, or iPod touch - Apple Support (IN)
If your iPhone, iPad or iPod touch won’t update
If your iPhone, iPad, or iPod touch won't update - Apple Support
You can do the update via PC, see this Apple article: Update software on iPhone, iPad or iPod touch in iTunes on PC
Update software on iPhone, iPad or iPod touch in iTunes on PC – Apple Support (UK)
If you have a Mac see this Apple article: Update your iPhone, iPad, or iPod touch with the Finder
Update your iPhone, iPad, or iPod touch with the Finder - Apple Support
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If the Screentime issue is not resolved after updating to iOS 16.4.1, visit an Apple Store or an Apple Authorized Reseller in your region for assistance to diagnose the problem.
And they could offer a possible solution.
You will first have to make an appointment due to the pandemic.
See: Contact Apple for support and service
Contact Apple for support and service - Apple Support