Combine apples, water, lime juice, and 1 1/2 teaspoons white sugar in a pot over medium heat. Stir until well combined. Add 1 teaspoon cinnamon and cook over medium-low heat, stirring occasionally, until apples are softened and water has evaporated, 15 to 20 minutes.
Combine flour, 1/3 cup white sugar, 1/4 cup plus 2 tablespoons oats, 1 teaspoon cinnamon, and salt in a bowl. Mix in butter until crumbles form.
Pour apple filling into a 9-inch baking dish. Sprinkle crumble mixture over the top.
Bake in the preheated oven until the top is golden brown, about 25 minutes.