The Apple Watch does not come with any physical port that you can use to connect it to a computer. While there is a hidden diagnostic port, it requires access to a special accessory, which for obvious reasons Apple does not sell to consumers.
A Redditor, however, seems to have gotten his hand on this accessory. The small accessory connects to the diagnostic port of the Apple Watch on one end and features a Lightning port on the other end.
The diagnostic pins are hidden by default and one apparently needs to push the small button on the top forward so that they are visible. This is presumably done to prevent the connector pins from getting damaged. With the Watch connected to the accessory, a Lightning cable connected to a computer needs to be plugged into the accessory so that it is recognised by the PC.
The connector is used by Apple for diagnostic purposes when one sends in their Apple Watch for repair. It can, however, also come in handy to downgrade the smartwatch from watchOS 3 beta to watchOS 2.2 provided one gets their hands on the required IPSW file (and the connector).
Thanks Erick121 for the tip!