Hi there,
I’m sorry to hear that you’re experiencing issues updating your cellular firmware on your Apple device running iOS 13.1. My first recommendation would be to update to iOS 13.1.1 which was released several days ago. This may or may not resolve your issue.
The troubleshooting you’ve already tried is excellent. What I would recommend next is 3 things: 1, connect your device to a computer running the latest version of iTunes. Let the device sync and at the beginning of the sync you should be prompted to complete a cellular settings update. If not, 2, I would partner with your wireless carrier to determine if there are any issues preventing you from being able to compete the update. It is important to remember that adequate wireless coverage is necessary to complete this. Also, adequate WiFi signal is needed so if you’ve got choppy coverage in the other or both areas that could be contributing to the issue. Your carrier may recommend getting a new SIM card as well.
Lastly, you may want to consider the possibility there is an issue with your software that’s preventing the cellular update from completing. What would be recommended is backing up your device, but factory wiping and setting up as new. Upon setting up as new you will not restore from a backup, but sign into iCloud to sync your photos contacts and other iCloud data. This will be a fresh start but the cellular service update should be able to complete. Also see the following:
https://support.apple.com/en-us/HT201270
Hope this helps!