How is the Apple Watch connected to the Internet?
From the "Apple Watch User Guide"


  Apple Watch uses Bluetooth® wireless technology to connect to its paired iPhone and uses the iPhone for many wireless functions. Apple Watch can’t configure new Wi-Fi networks on its own, but it can connect to Wi-Fi networks you’ve set up on the paired iPhone.
  
  If your Apple Watch and iPhone are on the same network but aren’t connected by Bluetooth, you can also do the following on Apple Watch without iPhone:
  
  
  Send and receive messages using iMessage 
  Send and receive Digital Touch messages
  Use Siri  
  


The watch is using the Wi-Fi settings from you paired iPhone to access the internet.  As stated above it can connect to networks you have set on the iPhone.  The Bluetooth will connect to the phone and pass data back and forth.