Write (or find existing) GitHub Action for fetching another repository at the same branch name

This issue has been migrated from [#10767](https://github.com/matrix-org/synapse/issues/10767).

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We used to rely on `$BUILDKITE_BRANCH`for this. GitHub's actions are finer-grained and [don't have a 1:1 replacement for this environment variable](https://docs.github.com/en/actions/reference/environment-variables#default-environment-variables).

https://github.com/matrix-org/synapse/pull/10160 uses a shell script to try and extract a branch name from various environment variables. Failing that, it falls back to a specific named mainline branch. I needed this in https://github.com/matrix-org/sytest/pull/1129. It'd be nice if we had an action for this rather than copying and pasting a shell script everywhere.

Raising this against synapse for now, in lieu of a place for generic matrix-org issues.