It's way harder than you think. Maybe impossible.
I'm going to tackle this from a different direction. What is the minimum age to meet each of the three requirements?
Language is kept
This is an easy one. Language is extremely useful, and children acquire it early. Probably any group of children above five years or so of age will keep some sort of language.
Tech level above stone age is maintained
This is where things get a bit tricky. What exactly is tech level above stone age? Do they have to actually be able to make their own tools, or can they simply re-purpose existing scrap. It's way easier to fashion an extra shovel into a passable pitchfork, or an extra pitchfork into a passable shovel, than to make either one from scratch.
So if your goal is to have them be able to salvage portions of existing objects for their own purposes, I doubt you need to increase the minimum age very much. Certainly by the time children are 7 or 8, they'll be familiar with the basic premises and purposes of basic implements like the hammer, the plow, the saw, the anvil, etc. Human ingenuity is a pretty powerful thing, so they should be able to survive for generations without ever even needing to know that metal ultimately comes from the ground.
If you want them to be able to produce new goods "from scratch", that's a lot trickier. Knowledge of smelting, for instance is almost certainly lost. Not only do very few children (or adults, for that matter) know how to recognize the proper ores and build sufficiently hot fires, but the bigger issue is, none of them will actually care. If any significant portion of the population disappears, there will be so much scrap lying around that for probably at least a generation or two it will be far easier to scrounge what you need than to make it yourself.
Minimum age for scrounging: 7-8
Minimum age for producing: 30-40 (The issue isn't that you need to be 30 in order to know how to smelt, it's that if you lose more than about half your population, it's going to be way easier to scrounge than to produce. There's plenty of extra lying around.)
Sufficient artifacts and knowledge from our current civilization survives, so, eventually, younger civilization can utilize it for a "reboot"
This depends wildly on what you mean by "reboot". Given enough time (say 10,000 years or so), any group of people will "reboot" society. We did it once, we can do it again. But I assume you want something faster. So let's say 100 years. What can our society do after 100 years?
Not much, it turns out. At the most optimistic, we're back into the Industrial Revolution -- maybe 1770's tech. More realistically, though, we don't even get that far.
Again, this is much less a function of age than of the sheer number of people left -- the more people, the better. Existing knowledge is almost immaterial, in some ways.
First of all, what is this society not going to be able to do? You can kiss computers goodbye, for instance. Now maybe you're about to object we built computers in less than 100 years, so surely this society, with its current knowledge, should be able to do it faster. But this is wrong.
Yes, we were able to create computers in less than a century. But we had a bunch of existing knowledge to draw on. Early computers, for instance, utilized vacuum tubes. Probably no one alive today of any age knows how to make them; almost certainly no single person knows how to mass-produce them; and even if someone did know how to mass-produce them, they wouldn't have the resources to build, run, or maintain the factory necessary to do so.
So computers are out. So are cell phones. So are regular phones, probably. How many people know how to build an old rotary phone? Even if someone did, your new society doesn't have the manpower to maintain and staff the switchboard network. When people are worried about getting enough food and water to survive the year, calling New York just doesn't seem like much of a priority.
Automobiles are similar. We discussed the issue of actually producing new goods above, but even if we ignore those difficulties, by the time people could reproduce the Model T there wouldn't be much call for a Model T. To begin with, cars run on gasoline, and your society surely doesn't have the ability to produce it. Not to mention that without any maintenance for several decades, much of your road system isn't in great shape.
Basically, if you're kicking your society back to before the Industrial Revolution, don't expect them to get back to where we are now in less than two or three centuries minimum. My guess is it will actually take them longer, because so much of the knowledge and security that people had during the Industrial Revolution is now gone. No one is working on reinventing the cotton gin when they're worried where their next meal is coming from. No one's even planting cotton.
If we lost more than half the working-age people, the resulting impact to society would be devastating. Starvation would be widespread. Society as a whole wouldn't recover for centuries, even with adults still around. With only children? Good luck with that. Depending on how many millions they lose over the first year or two, it might take them centuries just to breed their way back up to Industrial Revolution population levels.
Minimum age required to fully rebuild within 100 years: 40-50 (approximately 2/3 of the working population)
tl; dr: If you're writing Young Adult fiction, I'd say that 17-18 is your minimum age. Not because I think it would actually work, but the genre does encourage age-related suspension of belief. There have certainly been other similarly absurd scenarios that have gone on to widespread acceptance (cough Hunger Games cough).
On the other hand, if you're trying to make a serious, well-considered construction of what society would be like if all the adults suddenly disappeared...understand that probably involves massive casualty rates, especially during the first few years (but also increases in other types of mortality thereafter; expect infant mortality rates to skyrocket, for instance); a return to subsistence farming; and abandoning any technology much more complicated than the plow for anywhere from a few to several centuries.
Keeping language and basic societal structure is easy. Keeping advanced technology is nearly impossible.