Thank you… and sigh, that’s a difficult question. In Westeros, the penalty for treason is death. A queen sleeping with another man besides the king (never mind that he was her brother) and bearing his children, has committed treason and would bear a death sentence.
Now, does the act of treason apply to the children themselves? I’m not sure. But when Ned Stark was faced with this question, with the thought of telling Robert the truth, it brought to mind the dead Targaryen children that he had witnessed Tywin Lannister lay before the Iron Throne. So we can conclude that Ned thought Robert would have the children executed — and Robert had (technically) raised them.
And Stannis Baratheon is a just man, “notoriously without mercy”. He lived in King’s Landing when Myrcella and Tommen were born, so he must have known them, must have seen them grow up…. and yet he still referred to them as “bastards”, “abominations born of incest”. But had he won at the Blackwater, would he have shown mercy to the children, perhaps have done as Ned wished to and let them go into exile?
Well, despite the fact that Stannis is not a cruel man, and even says (in the TWOW preview chapter) that “the wind that blows exiles across the narrow sea seldom blows them back”, I’m afraid I don’t think he would. In ASOS, Davos II, Stannis must deal with his traitorous former Hand (who had wanted to marry Shireen to Tommen), and says, “It is law. Law, Davos. Not cruelty.” Furthermore, in that same chapter, he says, “For such crimes there must be justice. Starting with Cersei and her abominations… I mean to scour that court clean.”
But let’s consider a possibility — maybe Davos could have convinced him to be merciful? But as Davos wasn’t able to convince him to spare the life of Edric Storm, Stannis’s own nephew… then I don’t think he would have been able to do much for the lives of two children who bore no true relationship to Stannis at all.
So… unfortunately, I don’t think Tommen and Myrcella would have lasted long under Stannis’ reign. Though unlike Edric, they don’t bear any king’s blood… so perhaps they would have been safe from burning, at least. Maybe for them, Stannis would order a simple execution. For justice. For law. “It is not a question of wanting.”