Generally, each of the Commonwealth Nations with sufficient industrial capacity armed their own military forces.
The Brits were having trouble getting enough SMLE to their troops, so the Canadians turned to the Ross rifle circa 1912. Later, the Canadians had access to UK-built SMLE, and later were given the full Data Package. The Anzac forces also used domestic arms, if to common ammo.
The Arms were labeled as to their location of construction, where ever those places were. Specific markings were typically limited to the Unit Disk in the stock.
Really local items were just that. So, the Aussies built the Austen and similar sub-guns. The Canadians were meant to order a bunch of M2 Hyde SMGs, but they got Sten production underway about the time the M2 was cancelled for the M3 grease gun. Canadians had some quantity of Thompsons, if memory serves, which meshed with UK issue.
South Africa used the arms the Brits had sent them, which did not have specific marking (IIRC).
The factory probably tells more than anything else.