Hi Garth
Can't tell you much about the Buffaloes, although I can say that they aren't Freemasons (or therefore, 'Masonic'), albeit they are a fraternal organisation with similar charitable aims etc.
However, there were at least four Masonic lodges in Iraq [names and numbers: Baghdad 4022, Iraq 4471, Mesopotamia 3820, Babylonia 4326) during the 'British Mandate of Mesopotamia' (for further details see an outline of the history here) - I'm not a historian by any means, so cannot comment on the accuracy of that article, but it does explain broadly why the British were there. Just as in India and other areas of the globe that saw a great influx of the British (military and civil) populations, it follows that they will bring various influences with them, including their social structures, organisations and interests.
Just as there were some Masonic lodges there in that time, I am sure that there would also have been other organisations, such as the Buffaloes, Oddfellows, Foresters etc etc.
I know the lodges were working until 1963, but understand them to have ceased working since then, due to the religious/political changes that have occurred in the country since then. I assume it's the same for other orgaisations, too.
Hope that helps,