While Desmond is in Royal Scots bootcamp, a Sergeant-Major belittles him for his erratic behavior due to his flashes. ("The Constant")
During casting he was described as Foley. 40s - 50s, any ethnicity, British, physically fit, tough task master, unrelenting. A performance reminiscent of R. Lee Ermey in "Full Metal Jacket"...Nice co-star. impeccable British accent. 
Desmond addressed him as sir several times and received no rebuke for using the wrong mode of address. This implies that he is a Warrant Officer Class I - either a Company Sergeant-Major or Regimental Sergeant-Major. The job of "beasting" recruits (British Army term for putting soldiers through punishing physical exercises) would undoubtedly be done by a non-commissioned officer and the only such NCOs to be referred to as 'sir' are Warrant Officers Class I.
It is possible, given that we see no rank badges or a proper cap badge (and his lack of Scottish accent, which would be normal in a Scottish regiment), that he may hold a more specialized rank, such as a PT (physical training) Corps Sergeant-Major.
He was seen wearing a beret which is a headdress which is not worn by Scottish regiments, which wear either a balmoral or a glengarry