Goats undoubtedly existed in eighteenth century England. Since they ate anything, provided milk and a very sturdy skin, even the poorest of the rural poor aimed to have one. But perhaps because of their low status, they did not attract much attention from the Improvers. Anyway, there's not much said about goats until the late 1870's, when the possibility of goat hair in textile production was seriously investigated. As far as I know, no goats in Jane Austen, either. (Prove me wrong, somebody!)
Nevertheless, Shiels, in his exhaustive paintings of British livestock, came up with this life-size painting of two Welsh Goats, some time before 1842. It didn't make the fashionable print-books of his works, though. Pity.
My personal experience of English goats is that they are resourceful, rather grumpy around strangers, and give a good product for great cheese. Their coats are wonderfully silky, too.