Well… I guess you already know what is my first thought about this : the Polish election. 😁
Charles is likely to be elected instead of Henry as he is older and is not king of France in this timeline. Does he still die as in OTL ? If so, Henry may be elected after him. With Charles or Henry as king of Poland, Stephen Báthory stays prince of Transylvania and Sigismund Vasa stays king of Sweden.
That being said, let’s talk about Francis and Mary.
Francis’ survival changes the French Religious Wars. For the short time he reigned, Francis gave some concessions to non-rebellious Protestants (the Edict of Amboise and the Edict of Romorantin) but he did not hesitate to violently repress rebels.
After his death, Catherine de Medici became « gouvernante » of the kingdom (not formally regent) and awkwardly tried to use Protestants to counterbalance the influence of the Guises. She just managed to totally lose control of the situation.
With Francis alive, the situation is less ambiguous. The duke of Guise is de facto price minister. He and Francis are likely to severely punish any attempt of rebellion. The prince of Condé was already condemned to death. Likely, he is executed. Other troublemakers like the admiral of Coligny may end up like him.
Is it enough to prevent the disaster that occured in OTL ? Maybe. Or maybe not. It may also have the opposite effect and lead to a massive rebellion.
Scotland was virtually lost. Marie de Guise was dead and the Lords of the Congregation had taken power. Mary was still queen of Scotland but it was essentially symbolic : she had no power over Scotland.
Of course, Francis and Mary are unlikely to renounce to Scotland that easily. They would send French troops to Scotland as soon as they can and would search support from some loyalist clans still favorable to Mary. That means civil war in Scotland. If rebels win, I guess Lord Hamilton would usurp the throne and his eldest son would marry Elizabeth. If loyalists win, one of Mary’s uncles may become regent of Scotland.
Francis and Mary may find substantial support in Ireland. Irish Catholics would likely be unpleased to fall under French domination but, obviously, they would prefer that to English domination. An attempt of Francis and Mary to kick out the English from Ireland would probably receive massive approval from the locals.
However, I am unaware of the strength of English forces in Ireland and I don’t know how realistic it is to imagine Francis and Mary managing to take Ireland. If they do it, another one of Mary’s uncles may be regent of Ireland. (Yeah, she had many uncles.)
England is an entirely different matter. There is no way for English Protestants to accept the idea of a union with France and even English Catholics would dislike the idea.
As a result, if Elizabeth dies childless, I am pretty sure English Protestants would simply apply Henry VIII’s Third Succession Act and give the throne of England to some descendant of Mary Tudor the Elder. English Catholics may want to have Lord Darnley as a successor but I think he would have only very little chances.
Elizabeth having a child or not, Francis and Mary would still claim the throne of England but the only way for them to actually take power in England would be an invasion. And such a thing is always difficult. Maybe possible but difficult. Moreover, Philip II of Spain would do whatever he can to prevent such a union of crowns, even if it means England has to stay Protestant.