According to 18th-century clothing expert James L. Kochan in his book Don Troiani's Soldiers of the American Revolution, bearskin caps were introduced in the British army with the Royal Clothing Warrant of 1768, a transcript of which can be viewed here. There are numerous examples of bearskins that survived the American Revolution.

Fusilier regiments, Grenadier companies, Foot Guards and Drummers were officially issued with fur caps. However this were limited to parade at home or other instances when full dress uniform was required. They were rarely if ever worn on campaign, instead these units wore a shako like other units. Up until 1802 the the older fur caps were worn(looking something like this). In 1802 a new type was introduced, it was 16 inches high rounded at the top with a plate on the front, not dissimilar from can be seen today. However, they were still rarely worn on campaign or on active service.