Whisk lemon juice, oil, onion, garlic, wine, pepper, Worcestershire sauce, hot pepper sauce, celery salt, and salt together in a large glass or ceramic bowl. Add game hens and toss to evenly coat. Cover the bowl with plastic wrap and marinate in the refrigerator for at least 4 hours.

When ready to cook, preheat an outdoor grill for medium heat and lightly oil the grate.

Remove hens from marinade and shake off excess. Save remaining marinade for basting.

Cook hens on the preheated grill, brushing occasionally with remaining marinade, until no longer pink and the juices run clear, about 1 hour. An instant-read thermometer inserted into the thickest part of the thigh, near the bone, should read 165 degrees F (74 degrees C). Discard any unused marinade.