On a later date perhaps, the thrones of Britain and France could unite if the Stuart's or Valois die out, though it could cause a lot of trouble.
Thesprion, who posted in this topic has a timeline that concerns such a union, of which Mary Stuart plays a vital role: that is, she has a son with her first husband, François II. It's definitely worth reading.
In the scenario I've described, the thrones of Britain and France would likely not unite; certainly they may be on better terms in the end, but the only possible union would mean the extinction of the Stuart line. Even so, there would be claimants through female lines that would have a better claim.
Mary's children by Charles IX have no claim to the English throne; I suppose technically they do, but at least de facto, Mary would be ceding her claim to England to James and his heirs. If the Valois die out, they would be succeeded by the Bourbons, as France follows Salic Law and female cannot inherit.