After chasing it up with them, I had a further response from Abarth. They gave me a procedure to go through of turning things off/on disconnecting/connecting etc and now it works! I don't know if it was purely that process or whether they've tinkered with something on the backend too. Oddly though, if I put my VIN in here >>
https://www.driveuconnect.eu/en/check-your-vin it still tells me that "Uconnect LIVE services are not available on your vehicle”.
In case it helps anyone else, this is what they told me to do:
In order to activate the UConnect LIVE Services with your account, please be informed to proceed with the following steps:
1) Cancel from the HU (radio) all the paired phones (press setting button on the radio and select Cancel personal data)
2) Turn-off the radio and the vehicle.
3) Cancel from the smartphone, all the Bluetooth information regarding UConnect LIVE Services (smartphone settings >>Bluetooth)
4) de-install mobile app from the mobile.
5) Turn off the smartphone.
6) Turn –on the smartphone and reinstall UConnect Live app on the mobile phone.
7) Enable Bluetooth connection on the mobile phone.

Log in into the UConnect Live app trough your mail and password (username),inserting the flag on “remember me” bottom.
9) Turn-on radio and vehicle and proceed with the pairing between mobile phone and UConnect system (radio) pressing phone and follow the instructions.