Bring the water to a boil in a medium saucepan. Add the potatoes and a pinch of salt. Cook over moderately high heat until fork-tender, about 20 minutes. With a slotted spoon, transfer the potatoes to a plate and let cool, then halve or quarter any large potatoes.
Add 1 teaspoon of salt to the potato water and bring to a boil. Add the peas and cook over moderately high heat until tender, about 5 minutes. Meanwhile, in a large skillet, melt the butter in the olive oil. Add the leek and cook over low heat, stirring occasionally, until tender, about 5 minutes. Reserve 1/2 cup of the potato cooking water, then drain the peas and add them to the skillet.
Add the potatoes and the reserved cooking water to the skillet and bring to a simmer over moderate heat, stirring. Add the mint, chives and lemon juice and season with salt and pepper. Remove the skillet from the heat and stir in the watercress. Transfer the vegetables to a warmed bowl and serve.