First, to set the stage: a magical emperor had forcibly taken over the world, and destroyed pretty much all actual physical technology that would be any unreasonable stretch ahead of 1500s, but after 18 years, he is defeated, and the people are left with the concepts of today. They are left for a thousand years, and coming back, how far would they be?
This question has been asked a lot on this site
The questions I've personally been involved with:
- How many years would it take to reach an industrial revolution level of life from scratch?
- How far would technology have progressed for a 100 person tribe with knowledge of current technology in 15 years?
- How long would it take to create a Windows 1.0 capable machine from complete scratch?
- Creating a chemical industry from a medieval tech level without petroleum
- Speedrun to the moon in one lifetime?
Here's the simple reality that affects all these questions...
Infrastructure: Without the infrastructure, it doesn't matter how much you know. I'm an electrical engineer with BiCMOS design experience and I can tell you that if I suddenly arrived... if everybody I've ever known in my life relating to my BiCMOS experience suddenly arrived... in the 1500s, it would take exactly the same amount of time to get to building BiCMOS circuits because the infrastructure is missing. I'm talking about the buildings that build the screwdrivers that let you build the machines that build the machines that build the machines that build the machines that allow you to etch micron-wide metal on a silicon substrate. We'd all be dead before the second generation of buildings, and the thousands (and eventually millions) of people trained to work in those buildings, were built. That missing infrastructure and all the technology that comes from it isn't handwavable. It's the single biggest insurmountable problem.
Exponential Growth: 99.9% of all human technology was invented in the last 150 years. From 1501 it took 370+ years to build the infrastructure necessary just to get to the ancient and honorable beginnings of modern electricity. Knowing about electricity doesn't mean you have the ability to extrude hair-thin enamel-coated wire to make motors and generators anytime soon.
Inadequate Preservation: People forget. The technology to easily recover what's in all those heads doesn't exist in the 1500s. It doesn't really exist today, but what we have today could be made to work with a concerted effort. But in the 1500s people are scribbling as fast as they can with iron gall ink on scraps of crappy paper and hoping manual type setting and hand-carved etchings can keep up with what they're trying to write down before they forget incredibly important chunks of information (hopefully nobody makes a mistake!). And this doesn't address the lack of convenient distribution of knowledge. We take planetary instantaneous communication for granted today. I play real time games with people in Turkey! Back then, you can't even reach everybody, and who you can are only reached at the speed of wind or horse.
Short Lifespans: If everyone on the planet, millions of people, had 2000's era technical knowledge it doesn't change the fact that they need to get specific kinds of knowledge together amazingly quickly with all the necessary labor... and all the necessary logistical support... before enough info is lost to build just the first generation of buildings and equipment. Unfortunately, there are no supermarkets, no mass farms or animal factories... people are still planting with plows and horses. The majority of your knowledgeable people are actually very busy just keeping people fed. And none of this deals with the fact that lifespans are on average much shorter in the 1500s because no one has access to modern medicine. The second wave of the Black Death occurred in the 1500s. A great many of the people you're depending on to advance technology just died.
The Technology Pyramid: Finally, it's really important to realize that technology is a massive pyramid of knowledge, experience, innovation, and mistakes. I know how to design BiCMOS chips. I don't know how to make the tools that make the chips. I don't have the programming skills to write the software that lets me do it. I don't have the knowledge to build the buildings. I don't even have the knowledge to plant more than a garden. Your average individual is massively specialized. But that doesn't really matter. Because even if you had access to libraries that had all the books (they won't, but let's pretend they do) that can get you from building a fire by rubbing two sticks together to building BiCMOS circuits... most of those books would be destroyed for one reason or another long before you can use them. Remember, your lifespan is short. That means a ferocious amount of time must be spent training children who are needed in the fields to feed people.
The problem is you're working from the assumption that all knowledge can advance human society. The truth is, in the 1500s where not a single computer exists, everyone who went to school to become Computer Science majors are worthless. They have nothing to program. Everyone who went to school to make plastic or gasoline are worthless, because only the most basic chemistry can be done for centuries. Keep in mind that most of your population are lawyers, hair dressers, retailers... people who won't be capable of contributing to advancing technology at all. The vast majority of people can't help and of those remaining, they're missing all of the infrastructure necessary to allow them to contribute. I build BiCMOS circuits... but I only have a vague memory of how Berkeley built the first cat hair Bipolar transistor. And you need to pass through that point before you can do what I know how to do.
So, when you ask how far your society would get from, say, 1501 given modern day knowledge and a goal of 1,000 years?
With the suspension-of-disbelief exception of being 10-50 years quicker, they'll achieve 2501 technology on January 1, 2501.
Why do you care? It's your world. Change the rules! Ignore the Real World because the Real World is boring (or none of us would be here). Choose how quickly you want people to advance and go do it. Our opinion doesn't matter and you should not be constrained by the rules of the Real World.
Beyond our technology
Simply knowing that the modern world can be rebuilt would speed the process enormously. Instead of forging an uncertain path to an uncertain destination, now people would know guns and steam engines and automobiles are possible, and they would want them again, and devote way more economic resources towards making them than were devoted the first time.
Even with nothing else - with no specific knowledge of how to make anything - this means we would retrace our technological steps several times faster. Simply because we're devoting several times more resources to the task.
But in your scenario we do have specific knowledge, which would again speed up the process by another factor. What you need in the first generation, is a group of interested preservationists with scientific knowledge to write down, in permanent form, as much as they can. Create an oral tradition. Carve it into stone tablets. Write it on parchment. Make sure it is spread far and wide in forms that will last. If that happens, then returning to modern technology will be much quicker, because not only will we know it is possible, we will be able to save a great deal of research time and effort. We'll know exactly what works.
We still won't get there for many generations. At first most everyone is going to be occupied with basic subsistence, which they will be pretty bad at initially. But, assuming enough of the memories are preserved, a few generations down the line people will begin to rebuild.
The first draw will be metallurgy, for swords and plowshares. This can be done fairly immediately, since all you need is a hot furnace and the right ores. The location of these ores is something that could be recorded by the preservationists. Only two or three generations down the line, even to a lesser extent in the first generation, you could have people making metal tools from bronze and later iron.
If you can make metal tools, then making guns and steam engines and the printing press isn't too far off. It's just a matter of making the right pieces of metal in the right shapes, and the preservationists could leave tablets explaining in general what those shapes are and how to forge them. It's quite plausible we could have guns and steam engines and the printing press again within a hundred years. The first faction to achieve this would immediately dominate their region, and moreover they would know it would let them dominate their region, so there is a great incentive for them to do it.
From there, we'd already be approaching the industrial revolution. The main thing stopping us would be the lack of excess population to work in factories and assembly lines, because of everybody who starved in the first generation. But we'd already have tangible evidence that following the tablets of the preservationists leads to huge advantages, and we'd have steam tractors helping us produce more food per farmer, so it would just be a few generations to build up to previous levels.
Anyway, it shouldn't take more than a few hundred years to advance through the industrial revolution - faster than the first time. The modern age would soon follow. And then we'd have at least five hundred years left of the original thousand. So by the time the thousand years are up, we could be colonizing Mars.