The simple answer is yes, enlisted soldiers serving in the armies on the Western Front could receive promotions. It was a fairly common occurrence for privates to receive promotions to a non-commissioned rank and as the war dragged on, it became more common for non-commissioned officers to receive commissions. The attrition of the Western Front meant there was often a shortage trained commissioned officers and to fill gaps non-commissioned officers could be commissioned to fill the gaps. Despite the unfortunate circumstances surrounding the commissioning of these men, it often meant experienced soldiers were promoted to positions of command. I can only speak for The Australian Imperial Force regarding about attitudes towards promoting men from the ranks but by the end of the war it had become policy to fill all commissioned officer vacancies by commissioning s non-commissioned officer. By the end of the war, most Australian companies and battalions were commanded by men who had been promoted from the ranks.

Jeffrey Grey: A Military History of Australia Jeffrey Grey: The Australian Army