I somehow got the name of the actual operator using the following CLI commands:
qmicli -d /dev/cdc-wdm0 --nas-get-system-info
qmicli -d /dev/cdc-wdm1 --nas-get-system-info
From the respective otput of each I can tell my router is connected to:
cdc-wdm0 -> MCC=216, MNC=01
cdc-wdm1 -> MCC=216, MNC=30
Using the table here:
https://www.mcc-mnc.com/ I can translate the MCC - MNC pair to an actual operator name:
MCC=216, MNC=01 -> Hungary, Pannon/Telenor
MCC=216, MNC=30 -> Hungary, T-mobile/Magyar
Now please tell me there's an obvious and simple way to get the current operator name within WebUI!