Stuart,
If the flywheel turns over but the engine does not, then the flywheel is spinning on the crankshaft taper; problem number one. But the clutch adjusting nut NOT turning with it is baffling. It would make more sense if the clutch adjusting nut turned, but the flywheel nut that it threads in and out on to did not. Then the clutch adjusting nut would un-thread off the end of the flywheel nut, and drop off. I would think the clutch would start to slip before it got to that extreme.
Alternately the clutch adjusting nut AND the flywheel nut could turn with the flywheel, then the whole flywheel would fall off the crankshaft taper.
But if the flywheel turned, and the clutch adjusting nut did not (assuming the crankshaft nut is stationary as the engine is not turning over) then there is no relative motion between the clutch adjusting nut and the crankshaft nut, so it should not un-thread. The last possible combination, but improbable, all parts of the flywheel clutch turn BUT the clutch adjusting nut. However even if this were possible, it would cause the clutch adjusting nut to tighten, not loosen.
Loose flywheels were always the bane of Douglas. The flywheel needs to be a really good fit on the taper. If the taper in the flywheel bore or on the crankshaft are torn up, then some remedial machining is required. Otherwise one proceeds with lapping compound to lap the tapers into a mating fit. Good enough is when you can wring the flywheel onto the taper dry, then not be able to pull it straight off by hand without giving it a twist to break the fit.
The next thing to check is if the flywheel nut is tight. You are not bolting on the cylinder here, flywheel nuts have to be dead tight. Second, is the nut actually clamping the flywheel into the taper? If the flywheel sits too deeply on the taper (lapped too many times) the flywheel nut might be bottoming out on the threads before it fully bears upon the flywheel. Leaving out the thick washer (drive dog, with the two tabs) from under the flywheel nut could also cause the nut to run in beyond its normal position and so run short of threads. Or this washer itself could be at fault. Again, if the flywheel is sitting too deeply on the taper, the washer could be abutting the shoulder on the end of the crankshaft. It can also be installed backwards, but that should be readily apparent. In all these situations you could have the flywheel nut done up tightly, but it is only just nipping the flywheel onto the taper instead of clamping it on securely.
I have also once seen a flywheel that had a ridge at the small end of the bore formed by the end of the crankshaft (the flywheel is soft.) On placing this flywheel on a crankshaft other than the one it had been mated to, the flywheel got hung up on this ridge and behaved as if the two tapers were miss-matched angles.
-Doug