do you have another key to try as it would seem issue with recognizing transponder chip in key if not you could try disconnecting battery for 30 min, and checking battery is in good state of charge otherwise if it is not recognizing the transponder in key it would require taking to fiat to look at may just need to set up the keys again for immobilizer..check battery and do you have cover for battery so not been water damage to electrics in that area
If you had the electronic security code available from fiat over the counter for aprox £45 some stilo's can be started for emergency purpose by pressing throttle and entering this code if you do a search the procedure is in a thread somewhere, but you will need electronic code for any of above to try