In a blender, combine the coconut milk, sweetened condensed milk, and evaporated milk. Blend until smooth.
Add the rum, cocoa powder, vanilla extract, ground cinnamon, and ground nutmeg to the blender. Blend again until all ingredients are well combined and the mixture is smooth.
Taste the mixture and adjust the sweetness or cocoa flavor if desired, adding more condensed milk or cocoa powder if necessary.
If you prefer a smoother drink, strain the mixture through a fine mesh strainer into a mixing bowl or directly into your serving glasses to remove any lumps.
Pour the Chocolate Coquito into serving glasses and chill in the refrigerator for at least 2 hours before serving to allow the flavors to meld.
Before serving, give the drink a good stir, then garnish with shredded coconut and cinnamon sticks if desired.