TL;DR Unsuccessful initial update attempt, had to restore phone to factory settings, then
recover from backup. Very painful, time consuming, and worrying. Good luck to you.
We've had the same issue on two different phones, a 13 Pro and a 12 Pro. We were updating via iTunes on Windows. In each case the update takes the phone into update mode, the progress bar just starts to move, then the iTunes app exits and the device is stuck on the Apple logo with just a tiny bit of progress on the progress bar.
- We reset the device (quick up vol, quick down vol, hold on power button until your finger goes numb and finally you get the Apple logo) to get to recovery mode. Tried to update again. No luck.
- Tried with a different cable. Tried with a different USB port. Nothing worked. Always got stuck in the same place. Tried restore vs update. Nothing. Always stopped very early in the process.
- Tried a different computer. Didn't work initially.
- Rebooted second computer and again, restored rather than updated because at this point we just wanted the phone back to working state. That finally worked. The restore worked.
- We moved the phone back to the original computer with the backup and recovered the settings. Had to reauthorize the phone, prove we were the owners, then iTunes let us select the backup to recover from. Success. Had to redo some settings such as Face ID, but the phone was now at 18.3 and functional.
We had started to look at making an appointment at a Apple Store. But yeah - they use the phone as second factor when you try to sign in so that's not helpful when the phone is broken. So I guess you would have to call.
Luckily it didn't come to that for us.
I suspect the issue is a corrupted firmware image that only some people are "lucky" enough to get when trying to update. The successful recovery was done on a second computer that downloaded the firmware again. Could the first firmware image be corrupted? Maybe. Maybe there is a corrupted image out the in the Content Distribution Network and just a subset of us are getting unlucky enough to pull it initially?