EDIT: Apparently this argument has already been discussed (see e.g The infectiousness of nihilism by Will MacAskill). I will revise this argument and post a better version in the future.
Note: There are many forms of nihilism all with extensive literature behind them. While this method I'll be discussing can be applied to some other forms of nihilism too, I will be sticking to moral nihilism for this post.
When someone encounters a (moral) decision in life, their choice often depends on their personal philosophy. But if your dominant belief is nihilism it will tell you that it doesn't matter what you choose because there is no right or wrong answer. I say dominant belief because you can't be 100% sure that your theory is correct. This gives us some wiggle-room. Say a person is a very stout follower of nihilism with a certainty of 95%. This still leaves 5% for other theories that, while not as convincing as nihilism, have different degrees of persuasiveness to her. Let's say 95% nihilism, 3% classic utilitarianism and 2% various other theories.
At one point in her life the following event happens: There is a runaway trolley barreling down the railway tracks. Ahead, on the tracks, there are five people tied up and unable to move. The trolley is headed straight for them. She is standing next to a lever. If she pulls this lever, the trolley will switch to a different set of tracks. However, she notices that there is one person on this different set of tracks.
Now we can imagine the philosophies in her head as various voices that tell her what to do with various strengths. If we take the analogy of a parliament we can say that nihilism holds 95 of the 100 seats, utilitarianism has 3 seats and the various other philosophy share the remaining 2 seats. When she consults her philosophies what to do in this situation the dominant part (nihilism) will say that there are no right or wrong answers to this situation, so it's in effect not advising anything. Utilitarianism on the other hand is strongly in favor of pulling the switch. Since nihilism is withholding it's voice, the lever gets pulled.
So in effect, nihilism is entirely irrelevant here. And since it never prescribes a preference for what action you should take, you can always disregard it. A philosophy, which says that a certain category of decisions doesn't matter, doesn't matter itself when you have to make a decision in this category.