Disconnecting your personal iCloud account on your device is absolutely necessary before repair work can happen, to be taken back or resold. You can still do it via icloud.com even if your device doesn't work and/or is no longer with you!
Make sure you have made a complete backup of your device before following this step-by-step guide.
For your Apple Watch, follow this procedure provided by Apple Help Center: What to do before you sell, give away or trade in your Apple Watch, or buy one from someone else. If you no longer have the Apple Watch with you (or isn't functional), see this article: About Activation Lock on your Apple Watch