eggnog doubling the ingredients, what is the new cook time?
The goal is not to cook the eggnog for a particular time, but to a particular outcome, in this case to hit a temperature of 160°F, which will thicken the custard (eggnog is a very thin custard).

The exact amount of time this will take depends on the size and shape of your pot, and the heat output of your element or burner, and is hard to predict. 

You can use roughly double as a very rough estimate, but you will want to monitor the temperature, and remove it when it is actually done.