How can I unlock my Apple Watch Series 3 without my old phone?

You can reset your watch without the iPhone, have a look at the steps mentioned in this article:

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1. Put your Apple Watch on its charger, then press and hold the side button until the sliders appear.
2. Press and hold the Digital Crown to show the Erase all content and settings screen.
3. Tap Reset, then tap Reset again to confirm.
4. When the process finishes, set up your watch again. When asked, restore from a backup.

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copied from: If you forget your Apple Watch passcode - Apple Support

But after that, the Activation lock will still be enabled:

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If you can't access your iPhone or Apple Watch, or if your Apple Watch isn't paired with your iPhone and you need to turn off Activation Lock:

1. Go to iCloud.com and sign in with your Apple ID. 
2. From the Quick Access Menu in the top right, select Find My. You might be required to sign in.
3. Select All Devices, then click your Apple Watch.
4. Click Erase Apple Watch. Select Next until the device is erased.
5. Click the Remove button next to your Apple Watch.

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copied from: About Activation Lock on your Apple Watch - Apple Support