Combine apples, water, lime juice and 1 1/2 teaspoons white sugar in a saucepan 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.
Meanwhile, make the crumble
Combine flour, 1/3 cup white sugar, 1/4 cup plus 2 tablespoons oats, 1 teaspoon cinnamon and salt in a bowl. Rub in butter until crumbles form.
Spoon apple filling into a 22cm baking dish. Sprinkle crumble mixture over the top.
Bake in the preheated oven until the top is golden brown, about 25 minutes.
Lime juice can be substituted with the juice of 1/2 lemon.