This child between Henry and Anne would have eventually become King of England. This wouldn’t have changed Edward’s death, but it would have prevented Edward naming Jane Grey as his successor. Why name a woman successor when there was a male heir? It wouldn’t have made sense, and there is no way the council would have supported the idea. The country would have backed this second son all the way.
In an ideal world, the boy would have gone on to marry and have children of his own. The marriage would have likely been a Protestant one, and may have even been to Lady Jane Grey. It would have depended on the political situation at the time. His own children would have had Protestant marriages arrange for them, with daughters possibly moving to German and the Netherlands.
If the marriage was not to Lady Jane Grey, there is a possibility that a marriage to Mary, Queen of Scots may have been arranged. It would have been an important and advantageous marriage for an heir of Henry VIII. The question is whether Scotland would have agreed to this. At the time of Mary’s marriage to Francis of Valois, the country was still a Catholic one. Her councilmen would have wanted a Catholic marriage.
Mary may have still married Francis, but he died in 1560. That would have been when the English king was in his late teens/early 20s. It is possible that he was not married by this point and he would have become Mary’s second husband instead of Henry Stuart.