Alright, so I just bought a refurbished Apple Watch Series 3 with LTE and ran into the same issue when trying to pair it with my iPhone 13 Pro Max on iOS 15. Kept getting stuck in the loop of unable to update because you are not connected to the internet. I tried everyone's steps but hadn't had much luck. After a few hours and going out to eat to clear my mind, I decided to try a combination of everyone's suggestions and finally was able to get it to work on the first try! Here were my steps which literally was a combination of everyone's help:
I know the process is a little lengthy with multiple restarts and such, but I did this after hours of almost going insane and it finally worked, so hopefully it helps you too!
Hard Reset Apple Watch with the combination to erase all content and settings. This was turning off the watch by holding down the button under the crown. once off turn the apple watch back on and as soon as you see the apple logo, press and hold down the same button again (under the crown) for a good 15 to 30 seconds until you see the option to erase all content and settings. Do that and wait for the apple watch to turn back on.
Once back on just turn off your apple watch for good measure. This is just a regular shut down by holding down the button and sliding screen to power off.
Sign out of iCloud on iPhone
Hard Reset the phone by pressing on the volume up and let go, then quickly pressing volume down let go and then press the side lock button and hold down until the iPhone restarts.
Delete apple watch app on your phone
Hard reset iPhone again (with the combination of buttons)
Once the phone is back on, change your iPhone region (Settings ->General->Language and Region) to something else. I did Canada since I live in the US.
Hard Reset phone one last time
When phone is back on, turn on battery saver mode
Turn on apple watch
Download apple watch app back onto iPhone and start the pairing process.
In the pairing process make sure you also change the apple watch region to whatever you set your iPhone region to and also do the pairing process manually using the pairing code.
If you have the same luck, I did it should work, and the update should be on its way. Again, I know this process is lengthy and it requires multiple restarts and a combination of everyone's efforts, but this is what worked for me after hours of trying everything everyone suggested. If you get to a point that your about to throw in the towel, maybe try my lengthy solution and hopefully it helps you as well.