Preheat oven to 375 degrees F. Lightly spray 4 10-ounce ramekins or 1 9-inch x 5-inch loaf pan with cooking spray.
Cut 12 ounces baby bella mushrooms in half and add to the bowl of a food processor fitted with an S-blade. Secure lid and pulse 10 to 15 times until mushrooms are evenly chopped into small pieces. Transfer to a bowl and set aside.
Add 1 cup rolled oats and 1 cup walnuts to the food processor. Secure lid and pulse 15 to 20 times, until walnuts are ground and oats are chopped into a chunky flour consistency.
Heat 2 tablespoons olive oil in a large skillet over medium-high heat. Once heated, add diced onion and celery, cooking for 5 to 7 minutes, until onions are translucent. Reduce heat to medium and add minced garlic and chopped mushrooms to the skillet. Continue cooking an additional 8 to 10 minutes, until liquid from mushrooms is cooked out.
Season mixture with thyme, rosemary, sage, salt and pepper. Stir in tomato paste and Worcestershire sauce, cooking for another 1 to 2 minutes. Remove from heat.
In a small bowl, add ½ cup potato flakes and 1 cup hot water. Stir to combine and let sit for a few minutes to absorb liquid.
Whisk eggs in a large bowl. Add in ground walnut and oat mixture, along with cooked mushroom, onion and celery mixture. Fold to combine, adding in potatoes and cooked wild rice, until everything is evenly incorporated.
Divide the mixture into prepared ramekins or press into prepared loaf pan.
Make the glaze by adding ½ cup ketchup, 2 tablespoons yellow mustard, ¼ cup brown sugar, 1 teaspoon apple cider vinegar and ¼ teaspoon kosher salt into a small bowl. Whisk together until combined.
Brush the glaze on top of the Meatless Mushroom Loaf(s) and place on the middle rack of a preheated oven for 40 minutes, or until center is cooked to 165 degrees F and glazed top is set.
Let cool 10 minutes and serve with your favorite sides.