In this case, my version of Git had been updated by a regular system update, whereas my version of vim-fugitive had not been updated for some time.
This meant the new Git status format was not compatible with the older version of vim-fugitive's expectataions.
I use vundle to manage my vim plugins, so the solution in vim was to run
:BundleInstall!
This updated ALL my plugins of course, so hopefully won't cause more problems. But for now, my :Gstatus command in vim is working nicely again.
Thanks to Kamil Kisiel for this answer