Ok, lets say she gives birth to a boy and an girl, who are named Robert and Margaret, which are family names. She then leaves Scotland and the two little royals behind. Once she is stuck in England for what will eventually be revealed as the rest of her life, I can see her pushing for the children to join her. Mary Stuart was a quite maternal woman and she would probably push harder than OTL to see these children. They're not her direct heir/the new monarch and I can see a more direct struggle between than OTL.
Of the two, I see Robert being in more trouble than Margaret. He's probably given a title and eventually marred to someone who'll be helpful to his brothers regime. Perhaps he'd be put with Arabella Stuart, which might put him in the position to take the throne of England in lieu of his brother. If not, then a proper Scottish lass it will be.
Margaret will be married off to someone to make Scotland more politically strong. Perhaps Henri IV, after Margaret de Valois is set aside. Perhaps she would be married to Charles of Sweden or maybe John Casmir, Duke of Saxe-Coburg.
Ultimately, the twins don't really represent more than a chance for more foreign or domestic support with the added bonus of making the English succession more muddled.