I find the battery light comes on when it doesn't charge. Is the battery running flat or is it just the light not coming on?
Sounds like a diode has gone if it is losing charge. It's possible to replace the diodes, but I confess I don't know how. Alex will know, but he might be spending most of his time in the Stilo section!
An exchange alternator shouldn't cost a lot if the worst comes to the worst.
LOL - thanks Chas

Actually there's no reason for me to spend any time in the Stilo section, even though I'm driving it most days in the last two weeks, no maintenance you see

I get a false alarm almost every time I lock the Stilo with the keyfob button, so I lock it manually with the key (in the door) to avoid this. This has worked quite well for a week but now about half the time when I turn the key, nothing happens - but it's always OK on the second try. Wonder what will happen next week - maybe the false alarms will have stopped by then, maybe turning the key won't work at all. A little bit like Uno central locking, but without the 20-year-old motors and broken wires!
Now, back to the alternator topic.
You are indeed both correct, basically if the warning light never comes on, there's a good chance that the alternator won't be charging properly - it takes the initial current through the warning light to excite the windings in the alternator so that it can generate a current when the engine starts. That is why the no-charge warning light has a bigger bulb in it (3W wedge-bulb) than the other 1.2W warning light bulbs.
The current passes from the ignition switch through the warning light to the alternator via a thin wire pushed onto a small terminal near the much fatter main power cable. It's laughably haphazard - the most common fault is that the thin grey/yellow wire drops off the back of the alternator. The heat of the turbo nearby will probably bring it to a sorry end.
Then there's the possibility of corrosion within the crimped-on terminal, so re-make the connection from wire to terminal with soldering to be sure.
That counts for probably 90% of the no-warning-light scenarios.
Generally you have more of a problem when the warning light stays on. I've written screeds about the alternator and how to repair it, I'm going to do a couple of searches now and see if I can find some for you.
EDIT: try
https://www.fiatforum.com/uno-technical/98638-altenator.html?p=1063731
Sorry no pictures.
Check voltage across your battery with a multimeter. Engine not running, should be 11.5+ volts (ideally 12.30V after battery fully-charged with bench charger.) Then start engine and again check voltage across battery. In an ideal world this would now be 13.8-14.4V, but in reality as long as it's more than it was with the engine stopped (e.g. 12.8V), the alternator is charging. Just make sure that turning the headlights on and heated rear window does not cause the voltage to drop to what it was with the engine off. The alternator should produce peak output with the engine at 3000RPM.
Also check drivebelt tension/condition. Sometimes the belt slips without making a noise.
Cheers,
-Alex