Matt, seeing as you are looking for an urgent flight, don't mess around with what might be.
Contact all the carriers to see if any have some stand by flights and some of my friends travelled as couriers many years ago, I don't know if it is still done to London. But if you are ready to leave at short notice and any day during a weeks period, I think you can find something. Put your name down with as many carriers as you can.
Good luck. 
Mostly documents. Many companies wanted important documents hand delivered.
Probably not much of that anymore with so much going electronic.
