Bella55,
What source are you using to tell you he was with 66 Fd Coy? Clearly if this was his unit in 1915 and they were in Gallipoli, he must have been detached to the Mesopotamian campaign. But it is more likely that he was not with 66 Fd Coy at the time of his death. Unlike infantry soldiers who tend to stay with their regiment throughout their career, Royal Engineer soldiers can be moved between units quite regularly through what is known is trickle posting.
In August 1915 General Townshend's force in Mesopotamia was mainly the
6th (Poona) Division, which had only minimal sapper support in the form of 17 Coy and 22 Coy
3rd Sappers and Miners, who were Indian troops. It seems unlikely that a private soldier would have been detached to an Indian Engineer unit unless he had very specialised skills not found among the Indian troops.
August 1915 was actually a relatively quiet period in the
Mesopotamian campaign, between the battle of Amarah in the Spring and the
battle of Kut, operations for which began in September 1915. It is possible that he died of wounds he sustained in earlier fighting as the medical facilities were very poor and many soldiers who would otherwise have survived, died of diseases like dysentery. However since his grave has not been located, perhaps August is just an approximation of his date of death.
Unfortunately the War Diary for the 6th Poona Division does not appear to be at TNA. You may need to read some of the accounts of the campaign to get a better idea of where and how he died. Possibly start with General Townshend's own self-serving account: Townshend K.C.B., D.S.O, Major General Sir Charles.
My Campaign in Mesopotamia. 1920 London: Thornton Butterworth which you can find on the
Internet Archive. Lots of other useful references at the end of this article:
https://en.wikipedia.org/wiki/Charles_Townshend_(British_Army_officer)
Just a random thought. General Townshend was a former Royal Engineer officer and traditionally his batman would have come from his old Corps. Could Pte McReady have been on his personal staff?