The Tudors are a popular topic, and we often see points of departure discussing the survival of Arthur, Prince of Wales, so I thought I would propose something a little different.
In 1499 Henry VII fathered his third son, who he named Edmund after his father. Edmund was given the title Duke of Somerset at birth. A year after his birth, however, he died of natural causes.
So what if Edmund Tudor had survived to adulthood? Assuming that Arthur's death in 1502 is not butterflied away by the survival of Edmund, I think there are three interesting scenarios to consider arising from there being a second male Tudor line:
Scenario 1: When Henry VIII declares himself supreme head of the church in England in 1534, Edmund becomes the rallying point for a Catholic rising
Seemingly one of the failures of any concerted opposition to Henry VIII's reformation was the lack of a well-appointed Roman Catholic alternative. Had Edmund remained faithful to the Pope, he may have found himself seen as a Catholic alternative to Henry. Had this been the case (I find it a little unlikely) I don't see Edmund being successful, and he presumably loses his head, or at least spends a lot of time in the Tower.
Scenario 2: When Edward VI dies, Edmund is named as his heir
Assuming Edmund Tudor is still alive in 1553, and events go more or less as in our timeline, it seems likely that he is named as Edward VI's heir. It's quite probable that he would have still been alive, as both his father and his brother Henry lived to their 50s, and let's assume that he's also a Protestant of some description. If Edmund is dead by 1553, then we can substitute a hypothetical son.
Edmund would likely have been an influential figure during Edward VI's reign, and when it became clear that Edward was going to die he could have moved to secure his own succession.
Edmund would make sense as heir, far more than Lady Jane Grey did. First, Edmund is a man - at the time, the only previous experience there had been of a woman inheriting the throne had been Matilda, which hadn't been a positive time. Edmund would have been in a strong position, as a landowner, a protestant male, an adviser to Edward VI, and a son of Henry VII.
What could be problematic for Edmund is that this passes over Elizabeth in the succession. If she marries and has children, then these children could be challenge his family for the throne. His best bet, therefore, is to marry her to one of his sons.
Scenario 3: The Somersets inherit after Elizabeth dies
If Edmund and his family keep their heads down and don't rock the boat too much, when Elizabeth dies (assuming that the same pressures prevent her from marrying and having children as in our timeline) the crown passes to Edmund's line. Assuming he had sons, this means a surviving House of Tudor, for the time being at least, and also prevents the royal union of Scotland and England.
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As they say on exam papers: Discuss