Whisk Worcestershire sauce, soy sauce, lime juice, onion, red pepper flakes, dry mustard, salt, thyme, black pepper, garlic, and hot pepper sauce together in a bowl; pour into a resealable plastic bag. Add venison steaks, coat with the marinade, squeeze to remove excess air, and seal the bag. Marinate in the refrigerator at least 1 hour.

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Preheat an outdoor grill for medium-high heat and lightly oil the grate.

Remove venison from the marinade and shake off excess. Discard remaining marinade.

Cook the steaks until they are firm, hot in the center, and just turning from pink to grey, about 5 minutes per side. An instant-read thermometer inserted into the center should read 150 degrees F (65 degrees C).