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

[matrixbot]

This issue has been migrated from #10767.

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We used to rely on $BUILDKITE_BRANCHfor this. GitHub's actions are finer-grained and don't have a 1:1 replacement for this environment variable.

matrix-org/synapse#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 matrix-org/sytest#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.