Pretty much no one on either side would have worn wigs in battle.

Imagining battles during the American Revolution as stately, nearly coordinated set-piece affairs is one of the many disservices Hollywood has done to public understanding of the war. While soldiers on both sides did generally fight in formation in uniforms that seem impossibly bright and cumbersome by our standards, these were still human beings engaged in life or death struggles that punctuated long, arduous campaigns. Few people, officers or otherwise, would have had had the time to keep and put on a wig during a battle.

Depending on which battle or campaign you're covering, even the comparatively well-equipped British army was likely wearing cut-down uniforms modified for service in the American backcountry, Historical artist Don Troiani has created some excellent images informed by up-to-date scholarship and research of soldiers on campaign, such as these fellows from the Saratoga (1777) and Southern (1780-1) campaigns. You can see that formality has been exchanged for functionality. American units from the same campaigns, if anything, benefitted from operating from a shorter supply chain. Still, when marching about, no one but the highest ranking officers (who could store their baggage on wagons) would have the ability to keep a wig in wearable condition; you could imagine what even a single day of marching would do to the a wig if it was mushed into a soldier's knapsack.

Broadly speaking, wig-wearing was never as popular in the colonies as it was in Great Britain in the 18th century. The poorly funded and equipped Continental Army had a hard enough time supplying its men with the essentials of clothing and weaponry without commissioning thousands of wigs. On formal occasions, such as any ceremony or sentry duty where they might be seen by the public, enlisted soldiers would grease their hair back with tallow (rendered animal fat) and whiten it with flour. Officers might be able to afford talcum powder for the same purpose; this is what Washington did any time he needed to appear with whitened hair.

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For a good sense of the rigors of campaigning and the modifications soldiers made to uniforms and tactics when operating in the American interior, you can read With Zeal and with Bayonets Only, by Matthew Spring, or a touch less academically Patriot Battles, by Michael Stephenson.

The late, great Carolyn Cox explored how Continental Army officers sought to imitate the aristocratic trappings of European officers in A Proper Sense of Honor; this includes some discussion of hair and dress.