Preheat the oven to 350°F (180°C), or 160°C for a fan oven. Line a 9×5 inch (approx 23x13 cm) loaf pan with parchment paper.
In a medium bowl, mix together the flour, baking soda, baking powder, cinnamon, nutmeg, and salt. Put it aside.
In a large bowl, mix the softened butter and sugar for about 2 minutes.
Add the egg to the butter mixture and mix well. Add the rum and vanilla extracts and mix again.
Add the dry ingredients to the wet mixture. Mix until just combined. Add the eggnog and mix until smooth.
Pour the batter into the prepared loaf pan.
Put the pan in the oven and bake for 50-55 minutes. Check with a toothpick; it should come out clean when it's done. If the top is getting too brown, cover it with foil halfway through baking.
Let the bread cool in the pan for 10 minutes, then move it to a wire rack to finish cooling.
While the bread is cooling, mix the powdered sugar and eggnog in a medium bowl until smooth.
Put the cooled bread on a wire rack with parchment paper underneath. Pour the glaze over the top so it drips down the sides. Let it sit for 15 minutes before slicing.