Cook pasta according to package directions. When done, drain and set aside.
While pasta is cooking, wash potatoes and boil in salted water until tender (about 8-10 minutes). When done, allow potatoes to cool slightly and cut into medium-sized cubes.
While pasta and potatoes are boiling, sauté onions and peas together until cooked through.
In large mixing bowl, combine pasta, potatoes, peas, onions, and cucumber. Set aside.
In small mixing bowl, combine vegan mayo, olive oil, chopped dill, salt, and pepper and mix well to combine. Add mayo mixture to pasta mixture and stir gently to combine.
Serve immediately or cover and place in refrigerator to chill before serving.