"United Kingdoms" was actually the term used at the time James VI of Scotland also became James I of England. I'm not sure when the S was dropped,"

During the 17thC the Kingdoms of Scotland and England remained seperate kingdoms united in the sense that they shared a monarch. So in that sense the 's' makes sense. After 1707 the two kingdoms ceased to exist as seperate kingdoms when they united to form one united kingdom called Great Britain. Of course until 1801 Ireland remained as a seperate kingdom. At the time of Scottish/English union Wales was officially classed as part of the Kingdom of England itself hence the lack of Welsh representation on the Union Flag