Did you configure APN correctly?
I just tried for the first time and I think I managed to get data to work for a second after setting the APN and rebooted the phone. And then it did not work anymore after a minute…
Battery is now finished and I think I need to get SIM for testing so I don’t need to swap them back and forth.
I found one helper here:
https://developer.puri.sm/Librem5/Development_Environment/Boards/HowTo/Data_Connections.html?highlight=apn
But I Have not used these instructions myself.