472 years

This question is almost a duplicate of If all our technology disappeared, how long would it take to make a smart phone?, but I believe the loss of population makes it different enough. For the sake of future interest, I wanted the link to exist.

The estimated population in 2100 is 11.2 billion people. 5% of that is 560 million people, which is about the year 1650. So, if all we look at is population, and given that the average person today (in any country) has more basic knowledge than most people in the 1600s, we're reasonably assured that it would take 472 years to get back to the year 2122. (2122 - 1650...).

That's because the technology we have today is MASSIVE. Yes, the average person knows a lot and yes, the odds are reasonably good that masters of various technologies will remain. But, you can't run most modern factories with a handful of people — so a lot of tech is going to rot. In the end, I believe population growth will be the tall pole in the tent. Let's look at the details:

Advantages

• There is a LOT of raw material sitting around. From houses to factories to labs... a TON of material.




• There are a LOT of tools. From shovels to tractors to trains/planes/and automobiles. Wrenches and screwdrivers to last the world for a very long time.




• There are people who know what all that stuff (OK, 98% of that stuff...) is and knows something that can be done with them.




• Finally, human ingenuity is great. Nobody around who knows how to stamp metal? There are plenty of people on the planet who could find a metal stamping machine and figure it out. Reverse Engineering would be a wonderful pastime.

Disadvantages

• Chaos. A lot of chaos. Almost anyone living in a city has about two weeks to find food and shelter anywhere else because while every Walmart (or Walmart-like) store on the planet is sitting there with vast amounts of clothing and food, a substantial portion of that food will rot quickly and there's no more coming in. You'd think that, 95% of the population lost = 20X the food reserve for those remaining, except for the rot and the rats and the dogs and weather and everything else that will corrupt the supply. Small towns (say, 15,000 population) are fabulous for post-apocalypse survival. Cities are a death trap. And that assumes people haven't gone beserk and started killing one another to protect "their" personal Walmart.




• Weather will start eating away at everything. It's amazing how quickly unmaintained buildings (and everything they contain) fall into disrepair.




• Selfishness and panic (specific sub-components of chaos) will stop everything for a while. At worst, until the bullets run out.




• The power will go out. There's too few people to keep the electricity flowing in anything other than small local areas for a significant period of time. That means the cities go dark fairly quickly. However, there's enough smarts in the world that you'll see local wind farms, solar power (for as long as the panels last, there will be a lot to harvest), small hydroelectric... I suspect some communities will have electricity the whole time. Might even cause wars ("I want my MTV!")




• Finally zombies. Your question mentions nothing about zombies, but I like zombies, and what's a good post-apocalypse story without them? What, don't want zombies? Ah... party pooper! OK, no zombies.

So, what's the conclusion?

Technology

400 Years My gut tells me it's a wash, that the advantages won't balance the disadvantages. But my heart (ever optimistic) thinks that the advantages have an edge. But not a lot. I'm willing to shave off 2-3 generations of technological development. So I'm voting that you'll be back to 2122 technology by (from the perspective of 1600) 2050ish (on the your calendar: 2522).

Population

472 Years But you specifically said you wanted to return to 2122 population levels. Honestly, you really can't get people to procreate faster than they do now (or have done in the past). The boy-meets-girl-boom!-baby story has a fairly predictable clock. Granted, people enjoy better health today, live longer, yada-yada-yada, but the fertile birth period hasn't changed. Maybe it would be a bit faster, but I doubt it. So, like it or not, you need all 472 years to get back to 2122 population levels. Your calendar: 2594.

Final note: there are a considerable number of details that I'm ignoring. Like, what happens to nuclear power plants when not maintained over time (there's a question about that on the site) and what happens to hydroelectric dams when not maintained? What about random earthquakes, etc? What happens to cities when harbors/sea breaks wash away? Lack of maintenance casts a MASSIVE (second time I've done that) shadow on my numbers. But, so does all that human ingenuity. What if one of the 560M survivors was that Dyson dude or another clever inventor that can really boost local tech? What if the survivors aren't evenly distributed geographically, but are concentrated? There are so many completely unpredictable variables that my answer is, frankly, worthless. But it was fun to answer.

You can only assume that society would progress exactly the same if the variables are the same.

leaving it with only 1% of its current technology and only 5% of its population

This doesn't match any point in human history.

Technology generally follows a curved rate of development. With each previous discovery providing opportunity for greater discovery, but the rate of development seems to be correlated to population growth (from a history perspective).

Human population follows a similar curve.

I would argue that population size and density drive technology progression. If 5% of the population is spread out around the globe, than world wide progression of technology would be slowed, but if they are all in one continent then progression would be much faster.

Technology development will still follow a curve. If you need technology to develop to modern day levels, then don't take away so much at the beginning. Leave them in the electric age with the telegraph and steam engines of around the late 1800's. Keep the population tightly together and encourage population growth (provide lots of free farm lands). Things will get back to their screwed up state in no time. I would think just a couple hundred years.

It depends on which 5% survive, and exactly who's 2019 technology you're comparing.

1. We will also grow between now and then, at the time of this AI apocalypse, the world population will probably be in the 12-15 billion range, which puts us on 600 to 750 million people post apocalypse, if they decide that reestablishing the population to the approximately 8 billion we have now, it would take 100 to 200 years to get back to todays population levels. And as for what is in store for technology between now and then is anybody's guess, but if Moore's law holds for all technology then we'll still be more advanced after the 99% technology culling than we are right now in 2019.


2. There are still plenty of people in the world right now who still carry their water in buckets from the river to their home several times a day to get their water. If we only have to reach that level of technology, then I think that we'll be there already. If we instead have to reach the level of people flying around on private jets like some of today's billionaires, then we're going to take a lot longer.


3. Additionally if the survivors are concentrated in a more developed area, there would be correspondingly more survivors with the knowledge of science, engineering and technology, that they could teach to others and write new books, etc. But if the survivors are concentrated among the illiterate populations of the world, then that's going to stunt the knowledge retention of the race.

There's too little information to give a non-opinion answer, but probably never. If this level of civilization collapse occurred today, we still might not be able to restore ourselves to where we are now.

The main problem is energy supply. We've already harvested most of the easy to access resource deposits such as coal. If your AI destroys most of the computerized mining infrastructure, the fabrication facilities that create them, and deletes all the information required to rebuild it all, we're probably SOL.

One hundred and twenty years is really far into the future, but if current trends continue we will likely have consumed most fossil fuels and be supported mainly by renewable and nuclear power (of some sort). The AI would surely be able to sabotage most of that infrastructure, putting the survivors back at the wood burning and farming stage of civilization.

There might also be a huge amount of distrust in technology and anything resembling a computer, which would be understandable.

The time it takes to rebuild will be based mostly on that 5% of the human population left, and the traditions, rituals, and (ironically) religous ideas they pass on to their children. Your AI killed hard drives, not memories.

On Traditions (this may seem like a bit of tangent, but stay with me)

I remember reading, some years ago, about the Waste Isolation Pilot Plant - which is tasked to dispose of nuclear waste for 10,000 years. The WIPP did some interesting research on how to prevent the nuclear waste from being dug up in 10,000 years. We did up pyramids and tombs all the time, some of which leave inscriptions "if you disturb this tomb, then [such and such curse] will get you". Archaeologists laugh and keep digging. How can we explain to future civilizations that "no, really, if you dig this up a real, non-superstitious (but invisible) force will kill you. Slowly."

Their process involved a "do everything approach" - write down the warnings of radioactive waste in every known language. Leave large spikes in the ground as spikes seem to resemble "danger" to most animals. But, most importantly, they emphasized the need for traditions and educational material to emphasize the importance of radioactive waste. One of the authors cited the process Jews use to pass on Jewish traditions and rituals, which is interesting but makes sense. The Jews have been practicing their passover tradition almost the same way for 3,200 years. The importance of passing on traditions and rituals would be necessary to keep future generations aware of the dangers of this disposal site for thousands of years in the future.

Coming back to your question

The 5% of people left will remember what electricity, the internet, buildings, and technological advancements were like. They'll remember the pros and cons, remember the destruction of humanity by the AI, and so on. The two big questions are: (1) what will people tell their children? How will they ensure the information can be communicated to future generations?, and (2) How diversified / distributed are these people?

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  Imagination and engineering teaching children not only that things are possible - but actually existed can be compelling motivations for them to rediscover how things work. For example, imagine if your great grand-father remembered flying in a spaceshift that could travel faster than the speed of light. It would motivate us to not only believe it's possible, but we'd have a framework for talking about how - we can ask him what it looked like, felt like, etc. These would give us clues about the spaceship might work or function, which could potentially save decades of research into dead-end solutions. The same could work if suddenly we had no cars, skyscrapers, etc. Telling our children about elevators, airplanes, satellites, and the like could be compelling motivations to encourage them to pursue new technologies.


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  Making new writings If this 5% writes down everything they remember, information could be left for future generations to rebuild. Imagine if we found cave drawings of a manufacturing process for an element not on the periodic table. That would be a pretty damn compelling reason to figure out what it is! If the remnant of humanity just focuses on finding food (hunting) and they never talk about the "old world", then society would take thousands of years to rebuild. But if they're engaged and writing down their memories, even though they know they'll never experience that thing again, within a hundred years many of our technological inventions could be restored.


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  Population and resource distribution Milton Friedman talks about how complicated and connected the world had to be to create a simple pencil. If the 5% survived are all living in a somewhat central area, they don't have a chance in hell of mixing trees and plants from different continents to make a pencil. Our modern electronics involved plastics and metals from around the globe. If people can't simultaneously (a) get the resources and (b) communicate in a safe way, then rebuilding won't work.

Other answers have talked about how long it might take to get back to where we are at in population size, but this doesn't take into account that modern inventions don't necessitate a population of this size. It's the ability of people to perform the research (instead of hunting/gathering), have access to resources, and communicate with one another.

Jump-starting civilization with (a) the knowledge of what things could be simply because that's already happened, (b) some basic knowledge of the earth and resource capacity, and (c) a belief that people can work together - these are important factors. Knowledge of failed economic policies could save the rebuilding effort dozens of years. Knowledge that electricity is a thing would save us 100 years of trial and error on the idea. And so on and so forth.

However, the survivors won't live long enough (or have enough resources) to rebuild everything themselves. If the people who remember these things don't instill the knowledge of them as important traditions, writings, and beliefs in their children and children's children, then the information and knowledge will be quickly lost.