How do I pair a new iPhone with my Apple Watch?

Sorry, but only the previous owner can disable the Activation Lock for the Apple Watch, which can also be done via the iCloud Account over the Web:

About Activation Lock on your Apple Watch - Apple Support

Unless you have proof of purchase or ownership, as explained in this related article, the official Apple Support can not disable the Activation Lock:

Additional Support - Activation Lock

In case you can not unpair the watch from your old iPhone, follow these steps:

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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 Account. 
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

Then have a look at the steps to pair it with the new iPhone. If you restore your new iPhone from a recent backup of the old iPhone first, the latest backup for your watch will also be available after that:

How to pair your Apple Watch with a new iPhone - Apple Support