https://www.alternatehistory.com/Discussion/showthread.php?t=158353
It's an old thread I posted where I asked the same question.
Glass Onion said:
That is when would this alt Napoleon II be born?
Well, the issue of the absence of children was linked to Josephine's sterility. We don't know exactly how or why she became sterile, although there is a theroy that she suffered a miscarriage while being imprisonned. Anyway, with a POD that butterflies away Josephine's sterility, the child can be born as early as 1796 if your timing is perfect.
As a matter of fact, in the thread I gave, someone had mentionned Josephine might have thought herself pregnant before her marriage to Napoleon. Make this a fact, and you have young Napoleon II born in 1796.
Glass Onion said:
How would the presence of a male heir before 1810? How are the Napoleonic Wars impacted? What difference, if any does this change make?
The obvious consequence of Josephine giving birth of an ATL Napoleon II is that Napoleon I won't divorce her in 1809. I don't know if this is a good thing but to me that's a plus since Josephine and Napoleon were a very close couple: they even remain in good terms after their OTL divorce.
Josephine giving birth to Napoleon's son could also improve her relationship with the rest of the Bonaparte family who hated her and tried several times to convince Napoleon that he should divorce.
After that, it will depend on when this ATL Napoleon II is born. If we take the earliest option which is 1796, that would make him 16 in 1812. A funny fact that was brought up in the thread I mentionned earlier in the post was that it may lead to TTL Napoleon II marrying Marie Louise of Austria (or a Hapsburg relative) instead of his father.
I'm wondering if having a son earlier wouldn't affect Napoleon's personna and psychology. He might wish to give his son a stable and pacified Empire, leading to him being more cautious in politics and trying to avoid war. Yet again, I'm no expert on Psychology so I don't know how plausible this is.