Well, for one thing child rearing would become a much greater priority in our societies. Laws would be passed to encourage women to have children as quickly as possible. Single mothers would be given more financial support. Corporate practices which discriminate, even slightly, against women because of their fertility, would be pounced on as damaging to our society.
The entire education system would have to be revived. What happened to it in the interim years? Was it dismantled, since there were no children to teach? Or was society willing to pay to keep it up even if it wasn't providing value? If the former, it will have to be reinvented from the ground up.
Parents too old to bear children might be called into service to help raise them, so that those young enough to bear could have larger families. Couples too old to bear children could still raise them, so adoptions would be encouraged. The taboo against "paying" for children might be bent or even removed. Foreign adoptions would increase as fertile females in basement economic brackets see their fertility as a way to bear themselves out of abject poverty. Immigration laws would invite fertile females to make this move, and the countries they came from would try to keep them from leaving.
Nations which did not enact stronger laws protecting their women would find themselves shrinking. Note that this does not necessarily lead to good things for women, because women can be enslaved and turned into baby bearing machines "for the good of the country/state/whatever organization".
Babies may become a commodity. Child abductions may rise.
The first decade of the "post fertility cure" will have many middle aged women bearing children. "Use it before you lose it" becomes the slogan. This will lead to a high number of age related birth defects. This may lead to an emotional reaction against them. Women who don't get tested so they can abort any defective children will be considered irresponsible, or worse. Fertile wombs are a scarce resource and shouldn't be wasted on a child who is not going to be an asset to society. Unscrupulous nations may take "defective" children away from their parents and raise them to be "wombs for the nation", implanting them with fertilized eggs from "normal" parents.
Ten years later, another "dry spell" as the older women go into menopause and the world waits for the younger ones to grow up. Many countries may try to "force babies on the babies". Countries where children are married and give birth at a very young age will have an advantage over those where maturity is expected to happen later.
In any case, when the children of the "first regeneration" become old enough to be fertile, they will be encouraged to start having babies as soon as possible. By this time a strong societal acceptance for having younger women bear the children and older ones raise them will be in place. It may become a custom that children are most often raised by their grandparents.
Unwed teenage mothers will become society's "heroes" instead of its pariahs, as long as they realize their jobs is to have babies, not to raise them. "Your turn will come" becomes the slogan, featuring pictures of a happy middle aged couple being handed a baby by a proudly beaming teenaged mother, who then races off to her life of partying.
Society may also become paranoid about losing its fertility. Younger girls may be pressured to prove their fertility early. It happened once, after all. "Testing" young girls' fertility may be seen as an early warning system. A lot of what happens along those lines would depend on what caused the infertility in the first place. Was it man-made? And if so, was it deliberate or accidental? A natural disaster? Or do they simply not know what caused it?
--- further thoughts on children becoming a commodity / abductions ----
Nations will want to rebuilt their population as quickly as possible (lending a new meaning to the term "arms race"). Even though society hasn't completely disintegrated, the huge population decrease has left everyone vulnerable. What segments of society do you pluck from to plump up the others? What is your priority? Send people into the military or use them to educate the young? Nations run on their economies; who will work for the businesses that pay corporate taxes? Do we train people to build bridges and infrastructure, or to work on our electronic defenses? We can no longer have it all. hard choices must be made.
Or do they?
Nations with less scruples can find a way to game the system. Why pay all that money to raise and educate when you can purchase older children? Concentrate on your military and then you can pluck your neighbors, thereby having the best of both. Nations may specialize; imagine a country who acts as an educator and child raiser while being protected by the military of the nations whom they supply with trained workers. And smaller nations (Japan, perhaps) would concentrate on areas that require specialized training but not great numbers (the best hackers and anti-hackers in the world). Everyone knows that if you want the best programmers and analysts, you send your kids to the Nippon University of Electronics.
Once a child has been taken out of a country and provided with new identification it would be very difficult to locate that child without the help of the authorities. Of course, an obviously white baby in the hands of a black couple in Africa would be hard to hide, but consider how easy it would be to conceal a child stolen from black Americans or Asian Americans and transported to a country where their ethnicity is dominant. Or to conceal a child stolen from Africans and adopted by African Americans. Not to mention how easy it is to steal from your neighbors with the same ethnicity.
Then, imagine a wealthy man, woman or couple, too old to bear their own children and unable, for whatever reason, to adopt, or at least unable to adopt exactly the child they want. They have plenty of money and reason "we have so much money that any child would be better off with us than with someone poorer". Potential for abduction there. Wherever there is money there will be people eager to supply.
Then there are the baby makers. A new profession that grows up and flourishes in less developed countries willing to look the other way, as long as the profits are shared. Abduction would be an obvious supplement to the income that they get from selling babies. And for those worried about getting "caught" because of genetic testing, the children who are abducted could be used as "breeders" instead of being sold outright. Genetics could still connect them to a particular family but it couldn't be proved how those genetics got here.
These babymakers might pour money into genetic mapping and research so as to develop a menu for potential buyers. "Made to order" children with a carefully chosen set of genetic markers. Many of them might develop a very long range marketing strategy.