How to change the Apple ID used for my watch

Normally, the Apple Watch uses the Apple ID from the paired iPhone. to change it, you would need to change the Apple ID on the paired iPhone. It cannot have a different Apple ID to the paired iPhone unless it is setup through Family Setup for use by someone else that does not have an iPhone.

In which case, the watch would need to be restored and setup with the new Apple ID you want to use with it from the Family Group as outlined in the following link. Link-> Set up Apple Watch for a family member - Apple Support