Preheat the oven to 180 C / Gas mark 4. Grease a loaf tin.
Combine flour, caster sugar, brown sugar, bicarb, cinnamon and salt. Whisk eggs, bananas, oil and vanilla in a separate bowl. Stir into dry ingredients until just moistened. Fold in carrots and walnuts; pour into prepared tin.
Bake in the preheated oven until a skewer inserted into the centre comes out clean, 55 to 65 minutes. Cool for 10 minutes in the tin before removing to a wire rack to cool completely.