Put the potatoes in a medium saucepan and cover with cold water; season with salt. Bring to a boil and cook until fork-tender, about 15 minutes. Drain and rinse under cold water to cool, then transfer to a large bowl and add the celery.
Meanwhile, whisk the mayonnaise, mustard, olive oil, buttermilk, Worcestershire sauce, paprika and chives in a large bowl. Brush 3 tablespoons of the dressing all over the steak. Toss 1/2 cup of the dressing with the potatoes and celery; set aside the remaining dressing.
Sprinkle 1/2 teaspoon salt evenly in a large cast-iron skillet and heat over high heat until very hot, 5 minutes. Drizzle a small amount of olive oil (about 1/4 teaspoon) into the skillet. Add the steak; sear until well browned, 4 to 5 minutes. Reduce the heat to medium high; flip the steak and sear until well browned on the other side, 4 to 5 more minutes for medium rare. Remove to a cutting board and let rest 5 minutes, then thinly slice against the grain. Toss the watercress with the remaining dressing. Serve with the steak and potatoes.