Watch series 3 has been having problems with iOS 16 probably due to lack of storage on the Watch.
An iOS update is reported to be addressing this.
Make sure your iphone is fully up to date then erase all content and settings in the watch before trying to pair and update
Try these steps:
1. Make sure that your Apple Watch is correctly connected to the charger
If your Apple Watch won't charge or won't turn on – Apple Support (UK)
2. Restart your Apple Watch Click here > Restart your Apple Watch – Apple Support (UK)
3. Restart the paired iphone Click here > Restart your iPhone – Apple Support (UK)
4. Try again to update your Apple Watch.
If the update still won't start:
1. Open the Apple Watch app on your iPhone.
2. Tap General > Software Update.
3. Delete the update file.
4. Try again to update your Apple Watch.
if your Apple Watch says you need more available storage to install a watchOS update
.Click here > If you don’t have enough space to update your Apple Watch – Apple Support (UK)