Why won’t my iPhone 12 connect to my series 3 watch, and why does it keep saying I’m not connected to internet?

Series 3 watches will only pair to modern phones with some luck and a following wind. They are an old design (current model is series 8) but were sold up until last year which is regrettable.

Trouble Pairing, Watch no internet.

First check you have no other sports watches like Fitbit that are paired to your iPhone

Then force restart the watch. i.e. press and hold both the side button and Digital Crown for at least 10 seconds, then release both buttons when you see the Apple logo.

Then Force Restart the paired iphone   m

Force restart iPhone – Apple Support

Then in the iphone, Settings, General, Transfer or reset iphone, Reset, Reset network settings.

Make sure you have no VPN or VPN based adblocker

Then reset the router to factory settings which usually involves a paperclip on a little hole.

Then

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, restore from a backup.

Erasing all content and settings erases all media, data, and settings from your Apple Watch, but it does not remove Activation Lock if it has one.

Try again.

Then I recommend reviewing the steps in the articles below

How to pair an Apple Watch with your iPhone

 Click: How to pair your Apple Watch with a new iPhone - Apple Support or

Can’t pair an Apple Watch with your iPhone

Click:  If you can't pair an Apple Watch with your iPhone - Apple Support  or

Apple Watch not paired with your iPhone

Click:  If your Apple Watch isn't connected or paired with your iPhone - Apple Support