Place potatoes in a large saucepan and cover with water. Bring to a boil. Reduce heat; cover and cook for 15-20 minutes or until tender. Drain; cool slightly. Peel and cube potatoes.
Transfer to a large bowl.
In a small bowl, combine the vinegar, sugar, mustard seed, celery seed and 1/2 teaspoon salt; pour over warm potatoes and toss to coat. Cover and chill.
Just before serving, stir in the corned beef, cabbage and radishes if desired. In a small bowl, combine the mayonnaise, relish, onions, milk, mustard if desired and remaining salt; pour over salad and toss to coat.