Round the corner means nearby. However, I have difficulty with understanding the sentence "Round the corner comes the postman". Inversion has happened.
Kudos for recognizing that "(a)round the corner" can mean "nearby".
But sometimes things are to be read without second meaning: The postman simply walks around the corner.
"Round the corner" is not figuratively (=nearby) but quite literally used in this example. It follows the same pattern as
He walks "down the stairs" or "through the door".
If you are unsure which meaning is correct, you will have to check the context.
This is apparently from a book called Primary Education Volume 11 published in 1903 with lesson plans for elementary school children. A section on suggestions for February describes children making Valentine's Day cards for their classmates, which missives they seal in envelopes and give to a designated classmate to deliver, while they all sing
Round the corner comes the postman, Whistling on his merry way, While his happy smile is telling He has jolly times today.
The tune is that of the popular song "Clementine." The first line means "The postman comes around the corner," but the word "round" is chosen over "around" and the ordinary Subject-Verb order ("the postman comes") is reversed to Verb-Subject ("comes the postman") to fit the meter.