How to remove my Apple Watch old account

See this on how to erase your watch without having access to the paired iPhone

1. On your Apple Watch, tap Settings > General > Reset > Erase All Content and Settings.
2. Type your password if prompted.
3. For GPS + Cellular models, choose to keep or remove your mobile 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 might need to contact your network provider to cancel your mobile subscription. 

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

I just went through this with my wife's watch. In addition to resetting the watch, you have to sign into your Apple account and delete the watch there. If you have a new phone, then you set up the watch as if new.