How do I activate my Apple Watch without the original password?

You can reset your Apple Watch and set it up as new. You'll probably need the Apple Account information and password you used to set it up.

How to reset your Apple Watch if you forgot your passcode

1. Put your watch on its charger and keep it there until you complete these steps.
2. Press and hold the side button until you see the power button in the upper-right corner of the display.
3. Press and hold the Digital Crown until you see Erase all content and settings.
4. Tap Reset, then tap Reset again to confirm.
5. Wait for the process to finish, then set up your Apple Watch again. When asked, restore from a backup.

See If you forgot your Apple Watch passcode - Apple Support

Did you forget the numeric passcode you set or did you forget the password for the Apple account you used to set it up?