Theorically, it is possible to get a "Franco-British" Empire with Mary, Queen of Scots. Have Francis II of France survive longer, have a son with Mary (a daughter wouldn't inherit France) while she is still his bride, and then have Elizabeth die childless as per OTL. In this case, you get a personnal union of France, England and Scotland under Mary Stuart's son.
Technically, there are huge problems with that :
1°) England and France can't stand one another. Catherine de Medici only tried her friendly policies toward England AFTER Francis II's death. If the heir to the English throne is a french prince, Elizabeth is gonna be heavily pressurized into marrying.
2°)Religion. England, at the time of Elizabeth's reign, is Protestant with a persecuted Catholic minority. Scotland, though the protestant cause a little problem, is still Catholic. As for France, we are at the beginning of the religious wars : the Catholic are the majority, but the protestants are still troublesome.
A Union of the three kingdoms would lead to the eruption of a violent civil war. Not to mention that the Dukes of Guise, the most arch-catholic family of France, had a heavy influence on Francis II and Mary Stuart while they were ruling France : this CAN'T be good.
3°)Mary Stuart's position is very precarious in Scotland : the French alliance is starting to piss off the scottish nobles. A rebellion of the Earl of Arran, Mary's main scottish opponent, is very probable. It is also probable that Elizabeth is going to fund it.
4°) Foreign powers aren't going to like that situation, especially the Habsburgs. France and England are two of the major powers of the time : Spain and her Austrian allies are likely to wage war against the Triple Kingdom of France-England-Scotland if it ever comes to be.
To summarize and answer your question :
-The three countries could get united if Francis II and Mary Stuart have a son. However, it is likely that you're going to get a "War of the Three Kingdoms" which could turn out as a First World War...
-The English have a way out of this mess : Elizabeth marrying and giving birth to a child, and then have this child secure a bloodline. This way, even if Mary does have a son, this son won't be able to inherit England.