Does Ubuntu work on a Raspberry Pi Zero v1?

No, not on the Zero v1

The Raspberry Pi Zero runs the same architecture processor as the first RasPi, an ARMv6 arch ARM11. It's clocked faster than the original but it's not the ARM Cortex-A7 that the Raspberry Pi 2 has.

Ubuntu's ARM CPU support starts at the ARMv7 architecture.

You could recompile everything for ARMv6 but it literally would mean compiling and repackaging everything. It would be far more practical to just use Raspbian. This is a Debian redistribution that targets the original Raspberry Pi. Because it's Debian, it's technically very similar to Ubuntu.

Ubuntu's limited ARM support also means that Launchpad doesn't support ARMv6 so you won't be able to use Ubuntu PPAs either (unless they are naturally arch-independent, which some are).

Later hardware versions have different CPUs on, v2 can run 32bit Ubuntu currently, though that is not what this question deals with.

YES for Raspberry pi zero 2

From this page https://ubuntu.com/download/raspberry-pi

Ubuntu Server 21.10

Ubuntu core