I had a similar issue a few months back. I attempted to format a console with a bricked vWii (I didn't know about the bricked vWii at the time) and the Wii U format process basically has to boot into vWii mode to do a format of that mode. Since that would just crash, when I rebooted it I would be back at the format screen with the console saying it would now format the Wii mode, boot Wii mode, crash, repeat.
I was able to resolve it using UDPIH and the recovery menu. For this, you need a Raspberry Pi Pico or hacked Nintendo Switch in order to trigger the UDPIH exploit via USB.
Once in the recovery menu, I was able to modify /sys/proc/prefs/cafe.xml on the SLC, which is where the console stores its boot state (<initial_launch>). From what people have said on Discord, the possible values are:
0 - initial setup
2 - normal boot
3 - formatting Wii mode
255 - factory reset
There's probably more, but I don't know them.
Setting this to 0 allowed me to set up the Wii U as if it was fresh out of the box, so the console was working fine again except for the bricked vWii, which was fixed with an aggressive decaffeinate from Gary's vWii Decaffeinator.
Unlike in my case with the bricked vWii, it's not clear in this thread why the format is failing, so even if you're able to recover from the formatting loop via UDPIH, it's important to figure out how the console got into that state in the first place if you're going to be able to fix the problem. For that reason, it's worth also grabbing the logs from /sys/logs on the SLC.
I don't personally have the expertise to guide you through this process or understand the error logs if you're able to obtain them, but if you have or can get a way into the recovery menu, you should be a lot closer to diagnosing and possibly recovering from the issue. I'd suggest joining one of the Wii U homebrew Discord servers if you need more one-on-one advice on getting the recovery menu up and reading your logs. I was helped with my brick on Gary's Discord.
For setting the initial launch value, you can just use this fork of the recovery, which has an option for that: https://github.com/jan-hofmeier/recovery_menu/releases/tag/v0.5.0-3
That said there is a good chance that this is a failing eMMC, where the factory reset hangs, because it can't delete a folder.
The recovery also has a MLC Checker built in, which reads every file on the MLC to find corrupted ones.
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