How to set up iwatch series 3 without iphone

Ok a different question.

Series 3 are the oldest Watch that can pair with modern iPhones and have particular problems with iOS16. Make sure your phone is on 16.1.1 or later

Erase the Watch to free off maximum memory in the Series 3

On your Apple Watch, tap Settings > General > Reset > Erase All Content and Settings. 

Type your password if prompted.

For GPS + Cellular models, choose to keep or remove your mobile data plan.

• • If you want to pair your Apple Watch and iPhone again, keep your plan. 
  • If you don't want to pair your Apple Watch and iPhone again, remove your plan. If you aren't going to pair with a different watch or iPhone, you may need to contact your network provider to cancel your mobile data subscription. 

Tap Erase All to confirm. This resets your Apple Watch to the factory settings.

Now you can set up your Apple Watch again. 

I also recommend reviewing the steps in the articles below

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

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

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

Someone in the Family needs to have an iPhone

Set up your Apple Watch – Apple Support

Set up Apple Watch for a family member – Apple Support