I replaced the inductors in the Robot Shop wire following kit with the following:

Murata 10 mH ±10% Bobbin Inductor, Max SRF:200kHz, Q:48, 600mA Idc, 2.294Ω Rdc 1400

and swapped out the 22nF capacitor with one of 2.2nF. The results were much more uniform!

The new inductors have a lot less resistance and should be much more sensitive and be able to pick up the wire from a greater distance away from it. Maximum distance with reception was measured as being 300mm so I positioned the inductor bobbins at 150mm to get nice reliable results:

The screenshot below shows the wire positioned about 30mm to one side of centre:

… and the oscilloscope shows the red waveform being slightly higher than the blue. On the right hand panel is the Arduino serial out console and this shows a 'reasonable' steady differential of about 80 units out of a total range of 1024 units. This should give enough resolution for the machine to navigate to the required target of +- 5mm.