How can I reset the program on an Arduino Uno while the device isn't recognised by my computer at all?
I had the same problem with my Arduino Leonardo. But when I kept the reset button pressed and plugged it into my laptop, it got recognised again. But since you're using an arduino uno it might also be the atmega16u2 which isn't working properly. If holding the reset button while plugging in the arduino to the laptop doesn't work, try to reset your atmega16u2 chip on the arduino. This is the chip which makes the communication from USB possible. 

To reset the atmega16u2 follow this tutorial