As stated in the question, the branch name can be extracted from .git/HEAD and given [branch name], the commit ID can be found in .git/refs/heads/[branch name].
The package catchfile provides the command \CatchFileDef, which allows us to read .git/HEAD into a macro. As HEAD has no file extension, MiKTeX users have to add a trailing dot to the file name:
\CatchFileDef{\headfull}{.git/HEAD.}{}
This assigns something like ref: refs/heads/master to \headfull. As this string has a trailing whitespace character, we use \StrGobbleRight from the xstring package to trim it:1
\StrGobbleRight{\headfull}{1}[\head]
In order to extract only the branch name (master in the example) from this string, we can use \StrBehind:
\StrBehind[2]{\head}{/}[\branch]
This saves the branch name in \branch. Finally, we can use \CatchFileDef again, to save the commit ID in \commit:
\CatchFileDef{\commit}{.git/refs/heads/\branch.}{}
There are some edge cases where .git/refs/heads/\branch. does not exist: After running git pack-refs (which is a side effect of git gc --aggressive, for example), the heads are packed into the file .git/packed-refs instead of individual branchname files. As a workaround, check if the file exists before trying to read it:
\IfFileExists{.git/refs/heads/\branch.}{%
\CatchFileDef{\commit}{.git/refs/heads/\branch.}{}}{%
\newcommand{\commit}{\dots~(in \emph{packed-refs})}}
As a fallback, this creates the (not-so-useful) output "... (in packed-refs)" instead of a commit ID – but this only lasts until the next commit, when the file heads/branchname is recreated for the affected branch.
(A more ambitious workaround could parse packed-refs, of course.)
Full MWE:
\documentclass{article}
\usepackage{xstring}
\usepackage{catchfile}
\CatchFileDef{\headfull}{.git/HEAD.}{}
\StrGobbleRight{\headfull}{1}[\head]
\StrBehind[2]{\head}{/}[\branch]
\IfFileExists{.git/refs/heads/\branch.}{%
\CatchFileDef{\commit}{.git/refs/heads/\branch.}{}}{%
\newcommand{\commit}{\dots~(in \emph{packed-refs})}}
\begin{document}
This revision: \texttt{\commit} on branch \texttt{\branch}.
\end{document}
Sample output:
This revision: d92dc1386e48e04ceecb85461ed3b232146e6a32 on branch
master.
1 Using the last optional argument of \StrGobbleRight (name) to assign the trimmed string to a macro (\head) is necessary to allow further manipulation of the string using the xstring functions – see here for a discussion.