Well the guy at Tyre City in Halifax could not have been more helpful. He said without even looking at the car and going off his experience and the age of the vehicle, that it was almost certainly the alloy rim.
He popped the wheel off and showed us where it was fizzing through the edge of the rim at one spot. Then he wire-wooled it down and popped it back on, topped up all 4 tyres - all for no charge.
His advice was also to get the wheels refurbished and he recommended a place in Bradford that does a full refurb for £1 per inch - i.e. £15 for a 15 inch wheel.
http://www.cascoatings.com/As the car resides mostly in York, we will probably look for somewhere over there that can do this job and get all four wheels done, since one brand shiny new wheel on it's own would look odd, and at least one other wheel is showing similar tendencies.