https://www.alternatehistory.com/Discussion/showthread.php?t=157253
https://www.alternatehistory.com/Discussion/showthread.php?t=161176
These are two previous threads on which we talked about Francis II living longer or having a son. I followed the both of them because the first one was my very first post on this forum.
At first, I talked about the idea because in my mind, this scenario could end up with an Empire including France, a united Britain and Ireland (as it was an English possession at the time) under Francis II and Mary Stuart's son (which I had called Henry). However, the more I looked into the matter, the less plausible it seemed.
The first huge problem is religion. We are at the beginning of the Religious Wars between Catholics and Protestants. France and Scotland are both strong catholic nations, but the Protestants caused a mess in both places OTL. As for England, it is staunchly protestant by the time of Elizabeth's death. We still have Ireland who has a Catholic majority, but the Irish where under heavy persecutions from England.
The result of the union of the three kingdoms of France, Scotland and England is likely to result in a huge and bloody civil war between Catholics and Protestants. This is very likely as Francis II and Mary Stuart were influenced by the Guise family, archcatholics with there eyes set on the French crown.
The second huge problem is that the succession will not be accepted by everyone, and firstly by Elisabeth I of England, who was very smart and one of the greatest rulers of England.
While it's true that Elisabeth didn't want to execute Mary Stuart until it was proven she tried to kill her OTL, she didn't liked her cousin. If Mary Stuart's son turns out to be the next King of France, Elisabeth will certainly try to avert this succession. The two threads I prestend earlier ended up with considering the possibility of Elisabeth marrying so that her child inherits the crown.
Other people who won't accept this succession are probably the Hapsburg as it creates a powerful rival for them : England and France are two of the most powerful countries of the time. They will probably try everything they can to stop this from happening.
Last huge problem : Mary's position in Scotland. Because of the Guise, the Auld Alliance between France and Scotland had become less and less popular. Even if Mary becomes Queen of France, she is likely to lose her crown in Scotland due to a rebellion.
So I doubt we would see a "British French" Empire, or at least not without a huge Religious War of Succession.
What could be interesting though would be how OTL Henry III will act while in Poland. From what I read, he didn't liked the place and this is one of the main reasons he went back to France when his brothers (Francis II and Charles IX) died without issue.