Say she conceived in the late 1790s, right around the time of the Egyptian Campaign. I assume the child would have become heir to the Imperial throne upon Napoleon's ascension.
What would have happened to them following Napoleon's fall? Obviously they couldn't be locked up in an Austrian palace or anything of the sort, since no other royal family would have a 'claim' to them, so to speak. I could see them drinking poisoned wine courtesy of Bourbon assassins.