Removing the battery revealed a catch which when moved over with a screw driver allowed the top panel of the mouse (the flexible bifurcated ends of which flex to activate the mouse buttons) to be removed.
The scroll wheel (remarkably car wheel like) has spokes. I am guessing* that at the bottom a beam of IR light (because there is no visible light) passes through the spokes to a detector on the other side, and the interruption of the beam is used to detect motion of the mouse wheel.
The wheel was indeed quite dirty, and I noticed a foreign object, a thin strand of unidentifiable material at the side of the wheel which could indeed have blocked light from passing through the spokes of the wheel. An examination of the spokes with a magnifying glass showed they were dusty.
I gave the wheel a clean with a soft artist's paint brush, including the spokes, a wipe with an alcohol impregnated lens cloth and a blast with an air cleaner, through the spokes and underneath and to the sides.
The scroll wheel is now functioning normally.
So don't bin that mouse when the scroll wheel stops working - give it a clean!
* I was right: superuser.com, has the details, including how the detector knows which direction the wheel is turning.