Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes. Drain.
Heat olive oil in a large saucepan over medium-high heat; saute onion and celery in hot oil until tender, 5 to 7 minutes. Add flour, salt, and pepper to onion mixture and stir until well-combined.
Slowly whisk milk into onion mixture; bring to a boil, stirring constantly. Reduce heat to medium and simmer until thickened, about 10 minutes. Add potatoes, flaked salmon, peas, corn, and dill; cook until heated through, about 10 minutes. Stir Swiss cheese and Cheddar cheese into chowder until cheese is melted and heated through, about 5 minutes. Ladle chowder into bowls and garnish with green onion.
