I would say that it is John Clay who is the protagonist of "The Red-Headed League." The protagonist in a story is the person who initiates the action and keeps it going. He is not necessarily the hero or the viewpoint character. Nothing in this story would have happened if Clay had not wanted to steal the French gold stored in the bank's cellar. He found a nearby shopkeeper who was looking for an assistant and offered to come for half the usual wage. Clay started the so-called League of Red-Headed Men in order to get Wilson out from underfoot. Clay only appears for a moment when Holmes calls at the pawnshop to ask for directions to the Strand, but his activities drive the story, and he makes a complete appearance when he climbs out of the tunnel into the cellar and is apprehended. The French gold, of course, is the MacGuffin, or "bone of contention," and it is Sherlock Holmes who is the antagonist. Holmes is preventing Clay from getting the MacGuffin. Jabez Wilson is only a pawn in the game. Holmes and Clay have been in conflict for years. Heretofore it might have been Holmes who was the protagonist trying to track Clay down and have him hanged; but in this story it is Clay who initiates the conflict and keeps it going, just as he keeps tunneling until the great climactic finale.
The protagonist is a term that is used to describe the main character of a short story, who is normally opposed by the antagonist. In this short story, as with all of the Sherlock Holmes stories, it is clear that the protagonist is Sherlock Holmes himself. It is he, after all, who does all of the thinking and deducing to solve the mystery that is presented before him, and it is he who faces John Clay, the antagonist, at the end. Note what he himself says after the case has been solved and he is thanked by Mr Merryweather:
"I have had one or two little scores of my own to settle with Mr. John Clay," said Holmes. "I have been at some small expense over this matter, which I shall expect the bank to refund, but beyond that I am amply repaid by having had an experience which is in many ways unique, and by hearing the very remarkable narrative of the Red-headed League."
The way in which Holmes acknowledges that he has had some emnity with John Clay identifies Clay as the antagonist of this short story and himself as the protagonist. He has, after all, solved the case again and prevented a robbery from occurring. Even though the story, at the beginning at least, mostly involves Mr Wilson, it is clear that it is always Holmes who is acting to solve the mystery and bring about a happy conclusion, making him the protagonist.