W.Y. Carmen, "British Miltary Uniforms" (1957), pp.112-3, in a section titled "Grenadiers and Fusiliers 1799-1820", notes that the fur head-dress "was found to deteriorate rapidly on overseas serviced, especially in hot climates. Thus it was discouraged abroad and became an article for full dress wear at home." The shako was used on active service.

"In 1802 a new pattern fur cap was introduced.... [...] These caps, although not for the battlefield, were noted as being worn during the occupation of Paris when the French caricaturists did much to place on record the uniform of those days."

Carmen offers details of the head-dress, of course.

Cheers - Howie