The most pressing strategic demand for oil in the region for Britain came from the fact that the Royal Navy had largely converted its ships (and all of its most up-to-date ships) from coal to oil. This oil was supplied by the Anglo-Persian Oil Company (the forerunner of BP) from oil wells in independent Iran but that was shipped through the Ottoman port of Basra (technically, the Persian island of Abadan, but that border could hardly be a consolation with Basra practically in walking distance). As a result, within something like 24 hours of the breakout of hostilities a British Indian Expeditionary Force sent in advance in anticipation of war had landed south of Basra, which was captured within a matter of days.

I'd always thought that the mass oil reserves hadn't been dug up yet

This is correct. There was some speculative assumption that there would be oil in the region (Some of which was correct, as in Mosul. Some of which was very incorrect, as in the Standard Oil Company of New York (Socony, later Mobil) exploration efforts in Palestine).

That oil would be important really only dawned as a strategic consideration towards the end of the war, and then only really in regards to Mosul. Otherwise the territory under control had largely been negotiated in advance or didn't otherwise involve questions about oil.

So otherwise, no, it was not nearly as a significant a problem or concern for Britain, not least because other than Mosul (and as would be discovered much later Basra itself, for instance the discovery of the Majnoon field in the 70s) the vast majority of the territories involved-- the Hejaz, Palestine, Transjordan, Syria, Lebanon, Turkey itself -- don't have significant quantities of oil.

Edit: Just to clarify, by "Mosul" I mean the Ottoman province, not just the city, so the entirety of the modern Iraqi Kurdistan region and the area around Kirkuk, which are quite oil rich, as opposed to the vicinity of Mosul city, which as I understand it is far less so.