If you only want the Uno to control the connection, you don't need to feed any of the wires from the HDMI cable into the Arduino. Instead, you'd have the Uno control some appropraitely rated transistors (one for each wire) setup as simple on/off switches.
Edit: If you aren't that familiar with electronics this is the type of circuit I'm refering to, again you'd need one transistor per wire in the HDMI cable. Depending upon the exact type of transistor you use, it might be possible to use only one pin from the Arduino to control all the transistors. The only limiting factor would be the total current draw on that pin, but if the current drawn per transistor is less than 1.5 mA then it should be OK.