I had the problem too. After a few attempts at a soft reset, I contacted support. Since I had all the data in the cloud, I was also ready to use a hard reset or backup. And with that it worked.
Start iTunes and connect iPad Mini 5 to the computer
iPad restarted with Power + Home
At the Apple logo just hold Home button until the recovery mode is displayed
The computer should now recognize the iPad and display a message that an error was detected with the update and a new update must be loaded
iTunes loads iPadOS 15.0.2 - Again, an error was detected after the update was applied
iPad Restore - Last backup selected and installed
After the backup was set up again I had a working iPad Mini 5 with stable iPadOS 15.0.2.
Hope it helps you.