I understand that I'll need to change my tire. My question is how urgent is this? Can I just keep using the tire until it stops holding air, or is this dangerous, and the it might pop when I'm on the bike?
Seeing the cords in the tire casing is THE sign that the tire needs to be replaced. (Some tires have wear indicators, but they are overly cautious, so you should use the "do you see the cords?" test if you want maximal tire life.) It your case, you probably should have replaced the tire a bit earlier as the cords have been visible quite a lot of time judging from the size of the hole. But it's not too late.
The hole showing the cords is so large that you are living on borrowed time. The cords will wear. Eventually, the tire will burst, because the cords can no longer hold the pressure after being damaged, and the explosion will cause the tube to burst. A tube bursting in such a way cannot be patched easily.
I would not use the tire for anything except 1km shopping run, if following these procedures:
- Inflate the tire to its maximum pressure (let's say 7 bar)
- If the tire exploded, it's gone
- Reduce the pressure to a couple of bar below its maximum pressure (let's say 5 bar)
- Ride one kilometer
- Repeat
These will extend the safe life of the tire a kilometer at a time. It is unlikely that if the tire holds 7 bars of pressure before riding a kilometer, that the cords would wear so much that it can't hold 5 bars of pressure after riding a kilometer.
In my opinion, doing this safety check every kilometer is unacceptable, so it's best to simply replace the tire.
When replacing the tire, put the new tire to the front, and move the old front tire to the rear if it was the rear tire that was worn out. Front tires wear so little that the tire will die of old age instead of dying due to wear. You will always want to have the better tire in the front, or else the front tire wears so little it's at danger of disintegrating due to old age. A front tire blowout is much more dangerous than a rear tire blowout.