United States is plural because there are many states that have united,thats why.
United Kingdom is singular because there are 4 Nations united under one Kingdom.
Great Britain refers only to the countries that lay in the island of Britain,therefore we can not call Northern Ireland as part of Great Britain because is just not in the Island of Britain geographicaly,therefore it can only be part of the United Kingdom,which is made up by the island of Great Britain and the north of the island of Ireland.
United Kingdom ( England,Scotland,Wells,Northern Ireland) of Great Britain (the Island of great britain) and Northern Ireland,(like this we do not include any other indipendent country still conected to the crown since they are not part of the United Kingdom such as Australia,Canada etc).