Preheat the oven to 350°F (175°C). Grease and flour a 9×5 inch loaf pan with gluten free flour.
In a medium bowl, whisk together the gluten free all-purpose flour, baking soda, baking powder, salt, ground cinnamon, and ground nutmeg until well combined.
In a large bowl, combine the mashed bananas, shredded zucchini, granulated sugar, and brown sugar. Mix well.
Add the eggs, vegetable oil, and vanilla extract to the banana mixture. Whisk until smooth and fully incorporated.
Gradually add the dry ingredients to the wet ingredients, folding gently with a spatula until just combined. Do not overmix.
Pour the batter into the prepared loaf pan and smooth the top with a spatula.
Bake in the preheated oven for 50 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes.
Transfer the cake to a wire rack and allow it to cool completely before slicing and serving.