Ok, so I finally figured out how to reconnect my UX1 to my PC after disconnecting it to use it with my laptop.
When I plug it back in (usb connection), I get a red light on top of the UX1 - indicating that it's not working.
Rather than reboot, I did this:
Right-click 'My Computer' on the desktop, and choose 'Manage'. Select 'Device Manager' and right-click 'Sound, video and game controllers/Line 6 UX1'. Select 'Disable'. Wait until it disables. Right-click again and select 'Enable' - Hey Presto!
Well, it worked for me, anyway. Hope it works for you ;)