My friend submitted pics of the odd conical object he found with me on the 29th January to the friendly (and expert) folks of the MDF forum (http://www.metaldetectingforum.co.uk/). They identified it as a chape, or the metal point of a dagger scabbard. Quite a find! I like to think that the owner of the scabbard dropped it on the way to or from the building whose ruins I was investigating. Obviously the leather of the scabbard has long-since rotted away.
More pictures of the chape:
The chape would have looked something like this: