Not as far as I'm aware.
Due to the fact that the majority of the rural poor moved to the cities during the industrial revolution and became the urban poor, there isn't much of a tradition of proper rural poverty that the term 'Hillbilly' brings to mind.
The British rural tradition is much more associated with aristocratic landowners than the American one. The people who stayed in the country to farm the land were generally looked after by the local landowners in their country mansions, who built and maintained whole villages for them. To this day some of the most sought after and picturesque villages in the UK are ones that were built by the local aristocrat for his tennants, due to the fact that the village was a status symbol and treated as a fashion accessory to the country house.
Plus there's the fact that nowhere in Britain is far enough from 'civilization' to be truly deprived and isolated, except the Scottish highlands that gained none of these benefits and really did see extreme poverty and famine.
The majority of class related nicknames in the UK refer to the urban poor instead, names like chav and ned.
