In a blender, combine coconut milk, cream of coconut, evaporated milk, sweetened condensed milk, vanilla extract, ground cinnamon, and ground nutmeg. Blend until smooth.

Pour the blended mixture into a large pitcher. Stir in the white rum until well combined.

Cover the pitcher and refrigerate for at least 2 hours to allow the flavors to meld together.

Stir the coquito well before serving. Serve chilled in small glasses. Optionally, sprinkle a little extra cinnamon on top for garnish.