Warning: Contains Math
Let's start with the numbers side of things.
My city, Toronto, has roughly 3,000,000 people. A brief Google search shows its school board to serve roughly 247,000 students with an additional 91,000 in the Catholic school system. All totalled, that means that a bit over 11% of the population is serviced by the city's school boards. This does not get into adults catching up on their education, or any other schooling that might not be covered by these two boards.
For a population of 3 billion, you are looking at three orders of magnitude greater than this in total. Splitting the population evenly among the 16 burrows for convenience sake makes the average population of a burrow 187.5 million people. Yes, they aren't evenly split, but this is a starting point to discuss numbers
Using the (rounded) rate above of 11% in school, that make a student population of 20,625,000 split among all year groups in the average burrow. If the idea of 6 to 18 is to have 12 grades like the North American system, that amounts to 1,718,750 children per grade in this totally average burrow.
As pointed out above, they are not evenly split. Some are larger and will have more students. Some will be smaller and have fewer students. Regardless, it amounts to what we would consider a large city's worth of kids that need to be schooled.
Disclaimer: I will be using the system I was raised on, which was in Canada. Your experiences with schooling may vary
So, you have an average burrow school system that needs to teach 20,625,000 young impressionable children and you want to limit the number of places to send them. How do you do that?
The boarding schools for the kids are not schools at this point -- they are entire cities with the burrow. Not only do you have the 20+ million students in the city, but the million or so teachers providing instruction to them, along with the thousands of support staff for the facilities in the form of administrators, custodians, specialists, etc. With that many people, your school will need the infrastructure to support them on top of the regular needs for the school itself.
Side note: Not all students learn in the same way -- more resources may need to be added to handle that.
Even if you split these numbers into seven, it is still a mindbogglingly high number of adolescents to house in one place.
Then you have to figure out how to transport all those students back and forth on holidays and after the school session is over for the year -- where I live that is two weeks at the end of December and All of July and August. Assuming a schedule of five months in to one month out, that is still two round trips a year for students. A number that could increase with more, but shorter breaks.
The raw number of kids will probably lead to your children being split into a number of groups, say 12 groups spaced 4 weeks apart -- you're underground so you don't need to worry about years as much as if you were above it.
Even split in 12, that is still 1.7 million students that need to be transported in or sent home every four weeks on their schedule. Or more often if there are scheduled breaks in the year. Infrastructure that may not be heavily used outside those times. If at least one parent comes to pick up their children, that is roughly 3.4 million people on one transit route every four weeks. Your Boarding City also moonlights as a major transit hub or has a mini transit system of its very own to link with the burrow's system.
This part, your Burrow Government can do so long as they hold enough power to do so. This is making sure that there is a single lingua franca throughout the burrows and that your students are only taught in that language to a curriculum that they develop.
The only variance to the curriculum that is allowed is related to anything needed in relation to immediate geographical, and more importantly, geological area. Everything else your world government has set forth -- all of course curated by experts and containing enough subtle propaganda to help encourage children to love the empire.
Your world government's biggest challenge is ensuring that 16 disparate geographical area maintain a cultural homogeneity, even with the tunnels connecting them. Perhaps the apocalypse helped with that. But really your world government will want to ensure a steady flow of people from one burrow to another such that both the populations remain stable and there is enough mixing of the burrows to preserve a monoculture throughout the world. That is another logistical issue in and of itself.
Given the advancement of tech created these burrows, it should be plausible to house all the children in a small number of schools -- at least on a technical level.
Realistically though, I don't think it's feasible to have these boarding school cities within the burrows. Again, with the advancement of technology and a different mindset, I could be wrong.
Then again, I did not touch on the mental aspect of children going through 12 years of boarding school -- just the numbers and some of the logistics of supporting a student population of that size.