Place eggs in a medium pan and cover with cold water. Turn heat to medium and cook until the water comes to a rolling boil. Reduce heat and continue cooking for 10 minutes. Remove the eggs and transfer in a bowl of cold water to stop the cooking. When the eggs have cooled, peel them, quarter, and set aside.
Place potatoes in a saucepan and cover with cold water. Bring the water to a simmer over medium heat, season with salt, and cook until the potatoes are just tender and easily pierced with a fork. Remove the potatoes from the water and rinse briefly under cool water. Add the carrots to the pot of water and cook until just tender, about 2 minutes. Rinse under cold water to stop the cooking. Slice the potatoes into 1/4-inch thick rounds and place in a large bowl along with the carrots. Add the celery, radishes, and red onion and stir to combine.
Combine the buttermilk, sour cream, and mustard in a small bowl. Add to the bowl with the potato mixture and toss until potatoes and vegetables are coated. Refrigerate for at least 2 hours so that flavors blend. Before serving, season with salt and pepper, to taste. Fold in the watercress and dill and top with quartered eggs. Serve chilled or bring to room temperature before serving.