The mitre cap was adopted by the British Army sometime at the beginning of the eighteenth century. (Maybe late 17th. Orders regulating the appearance of the Mitre caps were not issued until 1743. The design established in 1743 remained virtually the same until the 1760s when it evolved into the first model of the bearskin cap. 1763 is the approximate date for the change over of the two (The end of Pontiac's Rebellion?)
As far as I know, the first British bearskin caps were mitre caps covered with bearskins to make their appearance more impressive. The first Royal Warrant regulating the bearskin caps was issued in 1768.