I don’t think they cut the cable.
The nest cam indoor / outdoor has an internal rechargeable battery. You can run it with the cable always attached in which case you have the opportunity to record continuous video or you can charge the battery periodically and use motion detection to initiate video recording.
I don’t have one in front of me right now, but the cable is a modified usb cable. One end is USB and the other attaches to the camera and only provides power to charge the battery (or to keep it charged).
Google sells replacement cables.
5 meter outdoor cable or this one that they ship with.
My guess is that they forgot to grab the camera before they left. If they realized they were intentionally leaving the camera, I bet they would have left you the cable to.