Such a child would certainly rise to become the King of Great Britain, but would he be elected as the next Stadtholder of the Dutch Republic?
My gut feeling would be to say, yes, he would become stadholder. Mind you not stadholder of the Dutch republic, but of the 5 provinces Willem III had been stadholder (Holland, Zeeland, Utrecht, Overijssel, Gelderland and the non-province of Drenthe). Friesland and Groningen will probably keep their own stadholder.
And would the personal union that began under William continue until this...
becomes this
in the modern era.
The United Kingdoms of England, The Netherlands, Scotland and Ireland.
I would say no for a variety of reasons. First of all it was never a personal union. A stadholder is not a king. Agood stadholder can be the most influential person in the Netherlands, but isn't the most powerful one. In the end, even with a strong stadholder, the Netherlands was ruled by the regent upper class.
Secondly, as I mentioned before, Willem III wasn't stadholder over the Netherlands. There was no such thing. You could only be stadholder over a province (and Drenthe). So Willem III had only been stadholder over 5 of the 7 provinces (and Drenthe). To be fair, he was stadholder over Holland, the only province that truly mattered, but still that doesn't mean he ruled the Netherlands.
Thirdly, the stadholdership isn't hereditary. Although I do believe the son of
Willem III would probably become tadholder, I doubt this would continue. The royal house would become more and more English (like what happened to the Welsh Tudors and the Scotish Stewards) and the Dutch, especialy the Hollanders would not accept it. After a couple of generations (and my guess it would be after the grandson of Willem III) they wouldn't accept an English stadholder anymore.
In that case there are three options, either they would ask for the Frisian stadholder to become stadholder of (some of) the other provinces, they would ask the second brother of the king, to continue the line, or they decide they don't need a stadholder at all. I believe this is unavoidable (mind you, I believe it is possible that some of the provinces, like Gelderland, keep the English king as stadholder, but Holland wouldn't). To be fair, it is possible that the English won't accept it, sent an army and simply conquer the Netherlands, forcing their king upon them. In that case you can get a Anglo-Dutch personal Union. Although France would love it and certainly get involved somehow.