Preheat oven to 350 degrees F (175 degrees C). Butter a fluted tube pan; coat bottom evenly with sugar and 1 tablespoon cinnamon.
Mix flour, baking powder, and salt together in a large bowl. Whisk in 1/4 cup butter. Add 1 cup milk, honey, and 2 teaspoons vanilla extract; mix until well blended. Mix in nectarines, 2 tablespoons cinnamon, and nutmeg. Pour batter into the prepared pan, patting and smoothing the top.
Bake until bread is golden brown, 20 to 25 minutes.
Place sweetened condensed milk, 1/4 cup butter, 1 teaspoon vanilla extract, and 1/2 teaspoon cinnamon in a bowl; mix until smooth. Mix in confectioners' sugar and more condensed milk if icing is too thick.
Cool bread in pan for 5 minutes; flip over onto a shallow, rimmed dish. Poke small holes into the bread with a fork; pour icing over the still-warm bread.