I can only speak for the Roman world - and briefly at that. From the first century BCE onward, many Romans seem to have carried a handkerchief-sized square of linen known alternately as a sudarium (sweat cloth) or orarium (face cloth). Although it had various functions - enthusiasts at the circus, for example, would wave their sudaria - its principal purpose seems to have wiping sweat from one's face.

At least sometimes, however, sudaria/oraria probably also served as handkerchiefs. One of Martial's epigrams mocks a man who was not allowed to blow his nose, and so managed to grow an icicle from its tip (7.37). It is, unfortunately, unclear whether he was desperate to blow his nose into a handkerchief or simply into his hand. There was, however, a relatively common expression describing a low-born man as someone who wiped his nose on his arm/sleeve, implying that a more refined personage would have expelled his mucus in a more seemly manner - presumably into a handkerchief. A final bit of evidence comes from the Christian apologist Arnobius, who describes a handkerchief unambiguously as a "mucinium" - i.e., a cloth for mucus.