Set the router to channel 10 or 11 and see if it shows up in the list of networks on the Pi. If it does, the problem is the radio properties and you just need to find the right country code for it to work (look at dmesg under Raspian and it should be logged).
You sir are a bloody legend!
I changed my router to channel 9, and my Wi-Fi network now shows up immediately! It doesn't seem to be effecting anything else, so this seems all good.
I can't thank you enough for all of your help. I would never have worked that out on my own.
Thank you again!