In the end, I got an "old school" Fiat mechanic to come round and look at it.
I explained again how it had a fully charged new battery, new sparking plugs and a new coil.
He asked me to turn it over - I did and it turned enthusiaistically without the slightest sign of wanting to fire.
He then asked for a screwdriver. I asked what type and he said it didn't really matter.
He pulled a plug lead off one plug and asked me to turn it over again. I did - and it fired almost instantly and started running on 3 cylinders
He then put the 4th lead back on and it ran as sweet as a nut.
In the cockpit, I hadn't been able to see exactly what he'd done, so I asked.
He gave me a sly look and said "Trade secret!"
Resisting the temptation to punch him, I persevered with my questioning.
He said that sometimes, when a car hasn't started for a while, it "needs a good run to earth" - and from what I'd been able to get out of him, he'd done this by shorting the screwdriver shaft (which was seemingly stuck into the plug lead) against the engine block.
My attempts with a plug lead
near the block to check the spark hadn't had a similar result, so maybe he really did just complete the circuit and run it to earth which somehow "cleared" the system. I can't explain it better and he didn't seem to want to, but now it runs fine again.
Make of all this what you will, but I reckon the new plug leads and coil were not even necessary. So if you ever have a similar problem, give this trick a try!
Bloody electrickery - utterly baffling!