Apple watch 3 unable to check for update

Series 3 is tricky

It will update to 8.7.1 if the iphone is fully updated and you erase the watch completely, but with 50% chance of re-pairing to ios16.2 iphone.

Ensure you have no VPN in the phone and reset the router to factory settings (not just a reboot)

erase the Apple Watch like this

1. Put your watch on its charger and keep it there until you complete these steps.
2. Press and hold the side button until you see the power button in the upper-right corner of the display.
3. Press and hold the Digital Crown until you see Erase all content and settings.
4. Tap Reset, then tap Reset again to confirm.
5. Wait for the process to finish, then if necessary set up your Apple Watch again. When asked, set up as new watch.