Hi David H.
Your measurement in-circuit of continuity from +D to ground is to be expected, it is simply the path back through the dynamo armature.
Those photos are useful. I think now that you have a Lucas CF3 cutout which incorporates an SB5 solenoid. This differs very slightly from the CF1 I had previously assumed in that the Fuse isn't in the main charging circuit but instead is used to supply auxiliary circuits. Austin didn't appear to use this feature, so the fuse does nothing and the bottom terminal in your photo (+ or +AUX) has no wire connected. Reading from the top, the terminals are A, Earth and +D
It is not impossible that there may be an internal fault in the cutout, for example a short circuit between the current (series) and voltage (shunt) coils, which are wound one on top of the other. Or the voltage winding, which has many turns of fine wire, may have gone open circuit. The outermost winding that you can see is the current coil, and the varnish doesn't look too good at one end. The heat generated in the cutout when it is running is pretty small but might be enough to detect by the "finger" test. I'd expect it to be warm but not hot after running a while.
If you want a more detailed description of how this particular unit works, take a look at:
http://www.da7c.co.uk/History%20Section/...%20CF3.pdf Remember to halve the quoted voltages for a 6volt one, so pull-in should happen at between 6.85 and 7.15 volts.
To help diagnose any fault, it would be useful if you could do some resistance measurements of the cutout with it isolated from the rest of the wiring, by removing the wires to the +D and A terminals. The earth can be left in place. Here is what I would expect to see:
Points Open +D to Earth 60 ohms
Points Open +D to A Open Circuit
Points Open A to Earth Open Circuit
Points Pushed Closed +D to Earth 60 ohms
Points Pushed Closed +D to A Nearly Zero ohms
Points Pushed Closed A to Earth 60 ohms
The 60 ohms is just my guess at the winding resistance of the voltage coil. I don't know what test facilities you have, the problem with a simple continuity tester is that any resistance below its threshold will register as continuity and anything above as no continuity. It will still give some indication though, so give it a go.
Over to you.... Cheers, John C.