Working in batches, puree watermelon (with seeds) and sugar in blender. Strain into a bowl through a fine sieve, pressing with a rubber spatula; discard solids. (You will have 5 1/2 cups juice.)
For children's drinks: Divide 2 1/2 cups juice among 4 ice-filled glasses.
For adults' drinks: Stir remaining 3 cups watermelon juice with tomato juice, hot sauce, Worcestershire sauce, salt, and celery seeds in a bowl set in a larger bowl of ice. Season with pepper. Stir in vodka and divide among 4 ice-filled glasses.
Garnish adult drinks with a piece of watermelon and a celery stalk.
To carve the watermelon, cut off the ends and stand it upright. Using a sharp knife, cut the rind from the flesh, making sure to remove all of the white. For a time-saving trick, puree the watermelon chunks -- seeds and all. Then strain the juice through a fine sieve. "It's easier and neater than picking seeds out," says chef Tony Esnault.