C.E. Franklin says that there were many types of fatigue caps, including stocking caps, but by 1810 the preferred forage cap for most of the cavalry was the type we see on these figures. Mostly blue, with a band that was white or in the facing colour. Other sources say the band was in the regiments metal colour, either white or yellow.
Infantry however preffered styles variously described as 'mitred', 'wedge shaped', or 'pork pie shaped'; as seen here:
says that originally fatigue caps were made from material left over when converting old coats into jackets so were red with a turned up front in the facing colour and looked much like this:
But by the time of the Napoleonic wars, coats were no longer being converted into jackets and red caps seemed to have gone out of fashion. Also the use and style of fatigue cap was not regulated but decided by each Regimental Command so there was an array of different styles and colours.
As mentioned above, cavalry adopted a forage cap which was styled on the Highland Hummel Bonnet and infantry preferred wedge shaped or stocking caps. But in 1812/13 the style of infantry fatigue caps was regulated. The new infantry caps were to be based on the cavalry forage caps (as seen on the Strelets and other figures) but some contemporary images suggest that they may have been stiffened to give more of a pill box or flared top look. There are suggestions that with regulation the colour of the caps might have been standardised, possibly grey or blue-grey.
There are also suggestions that caps of the Rifles Regiments might have been black or possibly green.
Re. These figures and The War of 1812, I believe what people are suggesting is to use the British infantry in fatigue caps as Highlanders. Some of our North American friends have told us that the Highland regiments found that kilts and ostrich feathers were not conducive to moving around the North American forests so started converting their kilts to trews, or wearing standard infantry overalls; and taking the feathers off their bonnets, which would make them look pretty much like a forage cap.