P0115 - the coolant temperature sensor. Which I have changed.
Did you check the sensor had the correct voltage supply/earth, or just replace it because the computer said so? Very often it's actually a wiring/connector issue that causes an error code- all the ECU can see is that it's not receiving the signals it expects, so flags up a fault- it doesn't know why.
Years ago I had a Panda with a similar issue- fan running constantly and a coolant temp sensor error logged. I could have assumed it was the sensor, but I checked the voltage at the connector- no supply. Traced the wire back to the ECU- still no supply. In a total random occurance, the output supply from the ECU had died.
From plugging in the diagnostic computer, to getting the wiring diagram and tracing the fault to the ECU- all in the 1 hour diagnostic we charged the customer. Granted , I know a lot of Techs who would've taken the easier option of just reading the code and condemning the sensor, but it always comes back and bites them eventually.
Now, you say you're getting the "Excess radiator temp" message- is this only on first start-up? If so, have a read of the common faults thread at the top (ECU update required, or at least, a decent battery). If it's coming up later on, then I'd be checking that your car isn't actually overheating/ has an airlock around the sensor/ has a failing headgasket.
You also say the gauge eventually starts working after a while/repeated start-ups- does it rise to the normal level? Is the cooling fan still running constantly? etc.
And finally,
https://www.fiatforum.com/grande-punto/339270-coolant-temperature-sensor.html
https://www.fiatforum.com/grande-punto/339215-replaced-map-sensor-coolant-temperature-sensor.html
You'll find an answer much quicker (and upset the mods less) if you keep it all in one thread.