While Pride and Prejudice was not published until 1813, two years before the end of the Napoleonic wars (1803-1815), Jane Austen actually began writing it in 1796, three years after France began warring with England (1793-1802) during the French Revolution. However, Austen probably continued revising the novel until 1805 and then again in 1811, during the time of the Napoleonic wars. In short, Austen wrote the novel at the time of England's wars with France during the French Revolution but revised it during the Napoleonic wars; therefore, it's not entirely clear which wars her subtle references to wars refer to, but they possibly refer to both wars.
Importantly, there are no direct references to either the war during the French Revolution or to the Napoleonic wars in Pride and Prejudice. However, the fact that a militia regiment has taken up residence in Meryton is a substantial clue that this novel is set during wartime and not during peacetime. The only reason for a militia to be stationed in Meryton would be to protect it. Meryton would have been located in Hertfordshire, which is just a little north of London. London is of course next to the English Channel, and just over the English Channel lies France, the country with which they were at war with. Meryton would not have been a significant town to protect, but it certainly would have been strategically practical to have several militia groups stationed in towns and counties just outside of London so that they can quickly move in on London should it have been attacked.
We learn for the first time that a militia has been stationed in Meryton when Austen speaks of Kitty and Lydia's need to frequently visit their Aunt Philip in Meryton to learn the latest news. Early in the novel, Kitty and Lydia return to Longbourn with the news of the militia, as we see in the lines:
At present, indeed, [Kitty and Lydia] were well supplied both with news and happiness by the recent arrival of a militia regiment in the neighbourhood; it was to remain the whole winter, and Meryton was the head quarters. (Ch.7)
In addition, by the end of the novel, Austen actually hints at peace being re-established, which is very interesting considering that she probably made her final revisions in 1811, four whole years before the Napoleonic wars actually ended. Therefore, we can probably conclude that Austen was referring to the war during the French Revolution rather than to the Napoleonic wars; otherwise she was predicting and alluding to the end of the Napoleonic wars four years before the end actually happened. The reference to peace can be found in the concluding chapter. Elizabeth has already long been married to Darcy and settled in Pemberley, and Lydia has long been married to Wickham. Austen mentions that Lydia frequently asks Elizabeth for money as she and Wickham frequently spend more than their income can afford. Lydia even continues asking Elizabeth for money after peace has settled in the country, which is when we see Austen allude to peacetime, as we see in the line, "Their manner of living, even when the restoration of peace dismissed them to a home, was unsettled in the extreme" (Ch. 61).
Hence we see that these two clues hint at a war between England and France, which is most likely the war during the French Revolution, although the book was written during the course of both French Revolution and the Napoleonic wars.