Running 16.1 my iPhone is not giving me the opportunity to update to 16.5

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

Yes