I’ve mentioned that there would have been more children, and likely more sons. Some of those children wouldn’t have lived long after birth, but that was the way of the Tudor period. I don’t think Henry VIII would have been as worried as he was with Catherine of Aragon or Anne Boleyn. He had his heir at least; proof for him that his marriage to Jane Seymour was valid.
There would have been at least one more son, I suspect. Henry VIII would have gotten that spare heir that he would had needed.
Prince Edward would have still died young, and it would have passed onto the spare heir to become King of England. However, that spare would have likely married and gone onto have children. England would never have joined with Scotland because of the necessity of James VI becoming England’s first King James.
It would have also meant that the scandal of Lady Jane Grey would never have happened. She would have remained somewhere in the line of succession, gone onto marry someone and had her own children. If Henry VIII’s line had died young without children, the crown would have passed onto his younger sister’s line as Henry wanted in the first place.
I do want to point out that I don't think there would have been many children after Edward. He may have had two or three siblings at the most that survived adulthood, especially if previous marriages were anything to go by.