It's possible there are examples of that usage but in general use, not as far as I'm aware. The usage "Middle East" caught on over "Near East" because the latter was understood to include the Balkans. By the time that the the Ottomans lost virtually all of their European possessions (i.e. by the outbreak of WWI) this was no longer seen as being relevant to the questions and policy issues of the "Middle East". Although there are earlier examples it was popularized in elite and policy making usage by Alfred Thayer Mahan in an article he published in 1902. This was then taken up by self-described "Middle East experts" like Mark Sykes in the pre-war years.

Nevertheless, parts of the Middle East were understood to be under the geographic purview of British India. So the Mesopotamian (Iraq) campaign in WWI was run out of India and fought largely by Indian troops.

The terms also remained somewhat fluid. There was a proposal during WWII to have Montgomery take over a "Near East" command of the Levant and North Africa while General Auchinleck would take over a "Middle East" command of Iraq and Persia. This was considered, but rejected.

Middle East is still something of a contentious term, particularly over questions of whether or not it should properly include North Africa, Sudan, Iran, or even AfPak. It's a bit jargon-y but I personally am perhaps most comfortable with MENA (Middle East and North Africa) extending west-to-east from Morocco and to Iran and north-to-south from Turkey to Sudan while restricting "Middle East", properly speaking, to these countries.