Would would have happened if the United Kingdom had been between England and the Netherlands and not between England and Scotland?
Its hard but probably not impossible to get this situation. The main problem is that the United Provinces, what is now the Kingdom of the Netherlands, was at least on paper a republic up unto the nineteenth century.
The likeliest way to do this is that William and Mary have a son. The Glorious Revolution happens much as IOTL. William of Orange persuades people in the United Provinces that they need a closer union with England to handle the French threat, and gets them to agree to a constitutional monarchy with himself or his son as the king. A union with England is negotiated.
The Scots don't go along with this, and in fact declare for the Old Pretender when William dies. Somehow this situation survives the War of the Spanish Succession. The French manage to do well enough to get an independent Scotland into the peace treaty. Setting things up this way obviously leads to lots of butterflies, and I'm not sure what happens to Ireland.