A charcoal grill gives the best flavor, but use whatever you have.
Prepare barbecue (medium-high heat).
Cut sweet potatoes into quarters. Arrange on large sheet of foil. Cover with more foil and seal edges to enclose. Wrap Yukon Gold potatoes in foil in single layer. Wrap purple potatoes in foil in single layer. Place foil packets on grill rack, 1 at a time, if necessary, and grill without turning until potatoes are tender, about 35 minutes.
Remove from grill. Open foil packets and cool potatoes completely. Cut all potatoes into 3/4- to 1-inch pieces. Whisk mayonnaise, lemon juice, mustard, and celery seeds in large bowl to blend. Add all potatoes and 1/2 cup green onions and toss to blend. Season generously with salt and pepper.
Can be prepared 6 hours ahead. Cover and refrigerate. Bring to room temperature before serving. Sprinkle salad with remaining 1 tablespoon green onions