How do I remove an old Apple ID account?

Actually the email provider no longer existing does not really matter. Just enter the email address as it was and update it at some later stage. If that is not working then you need to provide more detail such as what it says or does to make you think it is not working.

Without that account information (email address as it was and password) you can't reset the Watch. This support article has information about Activation Lock on your Apple Watch and how to turn it off. --> About Activation Lock on your Apple Watch - Apple Support

- Apple can completely reset a device if a person can provide sufficient evidence of valid ownership. Do this online at: Activation Lock support request - https://al-support.apple.com/#/additional-support Apple says, "Proof of ownership documentation is required," but does not state what currently constitutes proof. Prior to this new online service Apple typically required the original receipt issued by an Apple Authorized dealer and documented provenance (so typically it would be the original owner who does this, or a descendant with legal proof of ownership). Apple recommends you keep your receipt after you buy an Apple device. --> Keep your receipt after you buy an Apple device - Apple Support