You are right the Hanoverian troops were dressed in British uniforms. Line infantry uniforms for the line battalions and rifle style uniforms for the light battalions. The only differences were that some of the Hanoverians, especially the Landwehr were not as well equipped as their British counterparts and may have lacked some equipment, some Hanoverians may have worn plumes at the front of their shakos rather than to the side, the Landwehr wore stovepipe shakos, and the Grubenhagen light battalion had a peakless conical shako that looked like a mirliton. No need to swap heads, except to get the stovepipes, Hanoverians in schirmutz caps are almost certainly something that was invented by the 28mm wargaming community to differentiate the Hanoverians; I think it's a really good look and I always enjoy seeing Hanoverians in schirmutz caps but there's no real necessity to depict them that way because it's not actually correct.