This is almost certainly the novelette The Last Day of Christmas by David J. Lake.
I read this short story in the early 2000s in an anthology, but it wasn't new then (probably written during the Cold War).
The story is from 1981 and appeared in several anthologies, for example World's Best SF 1982.
In the story, the world was rapidly becoming overpopulated. ... demographic data wasn't being released.
The high birthrate and the talk about demographic data happens not at the beginning of the story, but around the turning point. Also at that point the relevant data was still freely released (mostly) and shows (also at that point) only a decline of the birthrate.
Only near the end of the story the friend (a historian who has contacts to the government) proclaims that there are no births anymore. At all. Anywhere on Earth. This of course is when even the western countries hide the relevant statistics ...
... some super-popular new cosmetic or perfume ...
In fact there are two new cosmetics (not counting the country-specific brands).
The first one is a perfume, the first aphrodiasiac that really works, and for a time it looks like humanity becomes more peaceful and happy. But the flaws of humanity reappear, the perfume is abused, and so the second cosmetic, a deodorant¸ is brought to market -- seemingly without any surprising effect.
... causing everyone to become sterile (and, I believe, also causing nuclear weapons not to work).
The combination of the active components of the two cosmetics causes both; what's more, the combined molecule also disables other gruesome weapons like Napalm and nerve gas.
... a company that turned out to be a front for aliens, ...
Only at the end of the story, at the eponymous Last Day of Christmas, the aliens reveal themselves and explain their secret agenda.
whether his newly-born daughter would be the last human alive?
The newborn daughter of the narrators friend (the one with the government contacts) is one of the last babies born. The narrator: "With the greatest of luck, little Eva could live to be the last woman on Earth"