OP must have disconnected the battery when renewing it so we can rule out a simple software reset. It does, however, smack of computer related issues. The two computers concerned are the Body Computer, located behind the fuseboard under the driver's dash, and the instrument (dash) control computer (which Fiat calls the NQS for some reason). The BC mainly controls all lighting and has software timers to reduce the number of external parts required. It also monitors lamp currents to be able to signal failures. The dash control computer does just that - controls the dash and all indication (I mean dash lamps, not direction indicators). These two computers are connected to each other via what's known as the CAN-Bus (a simple two wire system). There are other computers on the CAN-Bus as well (e.g. the electric power steering computer).
If looking at wiring diagrams in the Haynes manual, the BC is referred to as a"multi-timer", item 17.
Fault could be an earth issue, I would start at point E3, that is LH inner wing, near fusebox.
You could also remove the BC and inspect connections for corrosion/oxidation. Sometimes just unplugging and re-plugging can do the trick.
red.