I am not a historian or an academic, I am just a layman with an average job (though I do write in my spare time), but I would like to think to think of myself as a student of history, who enjoys reading history and acquiring historical knowledge as part of my own self-education. Middle Eastern history is one of my interests, partially because it is frequently in the news and some aspects of it is related to what I write about.
One aspect where my knowledge is rather thin is the history of British & French imperialism in the region after the fall of the Ottoman empire & the establishments of the modern nation states that currently make up the region. This period is very important because it does somewhat set the stage for much of the problems the Middle East faces today. I'm less interested on Israel/Palestine because I have read books on the Balfour Declaration and its aftermath, but more on the other countries like Iraq, Syria, Lebanon, Jordan, the Gulf states (even North African countries like Tunisia, Algeria & Libya would be satisfactory). My curiosity is based on wanting to know about the initial colonial occupations, the respective policies to these countries, how they 'managed' ethnic and confessional divisions, how did they deal with resistance, and later the support and influence over the regimes they erected after the occupations ended and the countries were formally granted independence.
I came across two books that looked interesting:
French Imperialism in Syria: 1927-1936 by Peter Shambrook (which is a study of French diplomatic papers related to the French imperial policies towards Syria. Looks fascinating as a piece of scholarship. I still want to read it, but not yet, as it looks rather deep for me at this stage, when I need to build up my knowledge and contextual understanding first)
Britain in Iraq: Contriving King and Country by Peter Sluggett
Are there any other books you would recommend on this aspect of Middle Eastern history, that are knowledgeable, but not too donnish and deep, at least for now?
You might be interested in The Guardians, by Susan Pedersen. It's a history of the League of Nations mandates system— the international system of imperial governance that the Allied powers created after the First World War. Under the mandates system, League members (eg, France, Britain) acted as 'trustees' for the various colonies they acquired from Germany and the Ottomans in the course of the war, while reporting to a mandates commission in Geneva that provided oversight and adjudicated disputes of sovereignty. Iraq, Syria, and Lebanon were all included in the mandates system, and the mandates system shaped every aspect of how Britain and France related to them, so Pedersen's book may give you quite a lot. She has a particularly great chapter on how the British manipulated the mandates system in order to contrive independence for Iraq—an 'independence' that just so happened to give Britain more, not less, control of Iraqi oilfields.
It's a dense book with lots of details, but far more readable than a pure monograph like the Shambrook you've cited. I think it might also give you some context for the Sluggett book, too.