Arduino UNO has a safe USB connection but CDC (unless you do some major hacking) not HID.
Jack Aubrey, do you have some way to go from COM: to joystick? If not then check out the Teensy 2.0 or Teensy++ 2.0 that are very easy to code as HID. If you're not a solder-Ace, get one *with pins* as those holes are real close to little surface-mount parts. The ones with-pins plug right into breadboards.
http://www.pjrc.com/teensy/index.html
They are a bit non-standard (Arduino compatible) but you can still use the IDE you have (Teensy software gives more options in Tools) and practically all the UNO code will run correct once you change pin numbers to match.
http://www.pjrc.com/teensy/teensyduino.html
Teensy 2.0 has 12 10-bit analog pins as opposed to the UNO's 6.