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After plugging the phone into a Windows conmputer (instead of my usual Linux), fastboot miraculously stayed on, and I was able to get the following information:

Doing that cache wipe did not solve the problem, unfortunately.

The Nexus 5 is unencrypted.

After plugging the phone into a Windows computer (instead of my usual Linux), fastboot miraculously stayed on, and I was able to get the following information:

Doing that cache wipe did not solve the problem, unfortunately: The OS animation has been running for 3 hours already.

Doing that cache wipe did not solve the problem, unfortunately.

Wiping cache

~ # twrp wipe cache
Formatting cache using make_ext4fs...
Failed to mount '/cache' (Invalid argument)
Done processing script file

~ # mke2fs -t ext4 /dev/block/mmcblk0p27
mke2fs 1.42.9 (28-Dec-2013)
Filesystem label=
OS type: Linux
Block size=4096 (log=2)
Fragment size=4096 (log=2)
Stride=0 blocks, Stripe width=0 blocks
44832 inodes, 179200 blocks
8960 blocks (5.00%) reserved for the super user
First data block=0
Maximum filesystem blocks=184549376
6 block groups
32768 blocks per group, 32768 fragments per group
7472 inodes per group
Superblock backups stored on blocks:
        32768, 98304, 163840

Allocating group tables: done
Writing inode tables: done
Creating journal (4096 blocks): done
Writing superblocks and filesystem accounting information: done

~ # twrp reboot
command is: 'reboot' and there is no value
Failed to mount '/data' (Invalid argument)
Failed to mount '/data' (Invalid argument)
E:a
Done processing script file