I recently heard this discussion, which wasn’t nearly as interesting as the question itself.
I suspect the answer is the US 50 Browning MG and its .50 caliber ammunition. This dates from the end of WWI and is still widely used by the US and many other countries. In the ordnance business, I can’t think of any other small arms weapon and cartridge combination that dates back that far. The 9x19mm cartridge dates back further, but not the weapon it was designed for. I can’t remember anything in the old Soviet inventory that goes back that far that is still in use today.
I have probably missed something. Would be interested in opinions!