ci: Bump `nightly` tag on scheduled Nightly builds (#18134)

This PR makes it so after a scheduled Nightly build we also update the
`nightly` tag to keep things in sync.

It's safe to bump the tag within this Action, as it won't trigger
another Nightly build due to GitHub's recursive Action protections:

> When you use the repository's `GITHUB_TOKEN` to perform tasks, events
triggered by the `GITHUB_TOKEN`, with the exception of
`workflow_dispatch` and `repository_dispatch`, will not create a new
workflow run. This prevents you from accidentally creating recursive
workflow runs. For example, if a workflow run pushes code using the
repository's `GITHUB_TOKEN`, a new workflow will not run even when the
repository contains a workflow configured to run when `push` events
occur.
>
> —
[source](https://docs.github.com/en/actions/security-for-github-actions/security-guides/automatic-token-authentication#using-the-github_token-in-a-workflow)

Release Notes:

- N/A