Bring a medium pot of water to a boil. Add whole potatoes and turn heat to a simmer. Cook until easily pierced with a fork.
Meanwhile, in a small bowl, combine the vinegar, mustard, garlic and mayo (if using). Season generously with salt and pepper, and whisk to combine. Whisk in the olive oil; then stir in onion and nuts.
Drain the potatoes and let cool completely. Using the same pot, bring about a 1-inch of water to a boil and steam the snow peas until bright and crisp, about 2-3 minutes.
Drain and run snow peas under cold water. Chop cooked potatoes into large cubes.
Toss potatoes, snow peas and turnips with dressing and adjust salt and pepper to taste. Let sit for 10 minutes before serving.