What if any evidence was there during the war of 1812, specifically on the North American continent what if any kit would the British officially appropriate from American stores, battlefields, raids, ect.
If so who would it go to native allies, militia, sold as surplus?
Not quite what you were describing but ships captured would find service on both sides.
In particular I will point out the USS President. She was one of the Original 6 Frigates of the USN. And of the larger 44 gun "Super" Frigate design. When captured by a British Squadron in 1815 off New York. HMS President would serve until broken up in 1818, and a new ship built along identical lines would be built and serve until being sold off in 1903