Where to go to get Apple Watch battery replaced?

Setup a service appointment at an Apple Store or Authorized service provider or ship it directly to Apple if you are in a supported country.

click here ➜ Apple Watch Service & Repair - Apple Support

From: https://support.apple.com/HT210551

"If your device performance has been impacted by an aged battery and you would like to get help with a replacement Apple Watch, contact Apple Support." Click here for Apple Support --> Contact - Official Apple Support