The back wheel on Bike.2 has developed a bit of a wobble; it's the bearings. Rather than submit it to the tender mercies of the Bike Shop, I decided to fix it myself. The method I chose was - replace the entire back wheel, plus gears and freewheel.
I did it this afternoon. I took off the old wheel, removed the tire and inner tube, inspected and cleaned them, and they seemed fine, put the same tire and inner tube on to the new wheel and put the new wheel onto the bike. It took me an hour or two, and it works fine now.
£25 for the wheel. No wobble!