Place the prepared potatoes in a large pot and cover with water. Bring to a boil over high heat, then reduce the heat to medium-low and simmer for about 10-15 minutes, or until the potatoes are fork-tender.
While the potatoes are cooking, prepare the dressing. In a small bowl, whisk together the olive oil, lemon juice or balsamic vinegar, mustard, minced garlic, dried oregano, salt, and black pepper until well combined. Toss the diced red onions into this dressing and let them sit. Set aside.
Drain the potatoes. Let them cool for just 3–5 minutes so they stop steaming, then toss them into the dressing while they are still warm. They will absorb the dressing.
Once the potatoes cool, gently fold in the cherry tomatoes, olives, herbs, and cucumber (if using). Taste and adjust the seasoning, adding more salt and pepper if needed.
Top with the crumbled vegan feta right before serving so it doesn't melt or get mushy. Chill for 1 hour before eating.