Engine cranks but does it crank nice and fast? The earth strap mentioned near the top of this thread is, on the Fiats, prone to corrosion and failure. On the Fiats it is under the battery box running from a gearbox bolt to the N/S chassis leg. You say the battery is "new" but is it well charged? If not really healthy the load the starter puts on it when cranking can drop the residual battery voltage to the point where it drops below the threshold voltage at which the ECU can operate. So you end up with an engine which cranks but plugs don't fire. However under these circumstances I'd expect cranking to be quite slow and laboured. Doesn't sound like this is your problem.
Edit. Crankshaft sensors do fail but often there is "history" leading up to the failure, typically acting up when hot. So miss fires with a hot engine or maybe reluctant to start when the engine is hot.