The problem with my bike is worse than my original diagnosis. The cause of the hole in the tube is a patch where the actual outer rubber portion of the tire has been worn through. As such, when you try to pump up a new tube inside, it bulges out of the hole. Clearly, the tire needs to be replaced.
That said, I only have three days left in Oxford, and nobody has committed to buying the bicycle yet. Regardless of what happens to it when I leave, the non-functionality of my bicycle now means that I need to walk to the John Radcliffe hospital in Headington for my final brain scan.
[Update: 29 June 2007] I have replaced the rear tire and the bike is now working properly. I am still looking for someone who wants to buy it.