Preheat oven to 350°F (175°C) with rack in the center position. Line a 9x5-inch loaf pan with parchment paper, letting it overhang the long sides by 2 inches, then lightly grease the parchment and exposed pan sides.
Peel, core, and dice the apples into ½-inch pieces. Toss them in a small bowl with 1 tablespoon of flour until evenly coated; set aside.
Peel the very ripe bananas and mash them with a fork or potato masher in a large bowl until mostly smooth with just a few small lumps. Measure out 1½ cups (340g).
Whisk the melted butter into the mashed bananas, then whisk in both sugars, eggs one at a time, vanilla extract, and sour cream until the mixture is smooth and uniform.
In a separate large bowl, whisk together the 2 cups of flour, baking soda, salt, cinnamon, and nutmeg (if using) for about 30 seconds until evenly distributed. Make a well in the center.
Pour the wet ingredients into the well in the dry ingredients and fold together with a rubber spatula just until no dry flour is visible — do not overmix; the batter should be thick and slightly lumpy.
Add the flour-coated apple pieces to the batter and gently fold them in with just a few strokes until evenly distributed.
Pour the batter into the prepared loaf pan and spread it evenly into the corners. Sprinkle turbinado sugar over the top if using.
Bake for 60–70 minutes, tenting loosely with foil around the 40-minute mark if the top is browning too quickly, until a toothpick inserted in the center comes out with just a few moist crumbs.
Let the bread cool in the pan on a wire rack for 15 minutes, then use the parchment overhang to lift it out and cool completely on the rack for about 1 hour before slicing.