How much would reasonably well-educated but unprepared time travelers from the present be able to speed up the technological and social development of civilization from the first empires (early bronze age) to the present level?
Or, to put the same question another way: How quickly could human society theoretically develop from bronze age to the present level? Historically we needed 4.5k years give or take a few, but we had a lot of wars, crises, collapses - surely we could do better.
Long version - story
The scientists told us not to go wandering off into that area. They warned us that if something went wrong with the experiment, any people in the area would end up in the distant past without any means to return. But it was too beautiful a morning not to go running.
And now, here we are. I have no idea where this actually is. People around here speak a language very different from ours; we learned a few words but communication in general is difficult. They say, the land is called Kemet and its ruler goes by the name Hor-Medjedu. Apparently, he is gathering laborers for some large building effort. His subjects seem to be quite chatty - if we could only understand them. Some of them are nice people, some are not, the same as in the world we will now never be able to return to. But honestly - I do not know how they can stand it to live here all their lives. The food is dismal (Imagine, no potatoes, no rice!), the weather is dreadful, there is no clean water, no electricity, no internet, everything is infested with disease, and deadly dangers are lurking everywhere. The other day, we all had to leave behind our shoes when we tried to cross this body of water before realizing there were crocodiles. At daytime, the heat is unbearable while nights are chilly. Soon, the locals say, the weather will turn wet and cold and large floods will rise from the river.
Who we are, you ask? Well, there is:
Alex - he pursued a degree in philosophy but dropped out to work as a carpenter and was recently unemployed, Jane - Alex' martial arts-interested girlfriend with considerable physical strength, a construction worker, Patrick - a gay PhD student in high energy physics - his colleagues were responsible for getting us into our current predicament, and Susan - Alex' mother, a well-established biochemist who always wanted to quit her job to travel the world but could never bring herself to do it. I guess, this time she succeeded.
A small group, to be sure. Now, if this was a fantasy novel, it would surely end up with Patrick marrying the king and governing by his side while Alex and Jane would rule as prince and princes over some other miraculous country and Susan would disperse her scientific wisdom as a motherly teacher to the delight of the common people.
Yet, reality is somewhat more disappointing. We have been told to avoid the local militia as they are said to be ruthless, undisciplined and have been known to enslave unsuspecting travellers. Apparently they are armed with stone clubs, wooden lances, bow and arrows as well as daggers made of a metal that is not iron, perhaps bronze. I have never seen any iron or steel around here nor any other metal that is common in modern age. Lucky for us. Until now, we've been trading the few everyday objects we had with us when we were stranded here for shelter and food. We present someone with a Dollar coin, they will look at it closely, hold it up against the sun, show it to their neighbors. Then, they will perhaps ask us into their house. We give them a bright lime-colored t-shirt and they will give us something to eat. It's not like bartering at all, it's more like mutual gifts. And we are quickly running out of gifts the locals would find interesting.
What will we do? I do not know. I still can't quite believe it. But it's real, incredibly real, brutally real. Will I ever be able to enjoy a decent cocktail again? Will I ever again have the luxury of indoor plumbing? Or of plumbing in general for that matter? I know how it works, broadly speaking. It's not difficult to build. We just have to get the locals to help us with it. They will love it. They just don't know it yet. What about electricity? I saw some copper and bronze items now and then, even some silver - that should make perfectly adequate electrical wiring. How we could make everyone's tasks so much easier! And then the rampant disease - how hard can it be to find some peniciline-producing fungus? If the locals knew how easy and convenient their lives could be! At present, however, they do not seem overly excited about our wisdom. That farm women got angry at us the other day when we tried to explain logarithms to her. Perhaps we would be more successful if one of us would be a little more aggressive and bombastic like Aleixo Garcia or certain presidential candidates?
Long version - out of character
It was pointed out in various other contexts that the disappearance of all technological achievements would make it very difficult to rebuild anything. Fine, you know in broad terms how an electric generator works, a computer, a radio, but you have to figure out the details and you will not find any ready made parts to construct those things from. More worryingly, the same is true for less advanced technology: Steel? A functioning printing press? An oceangoing ship? And still worse, while you might find descendents of formerly domesticated species in post-apocalyptic settings, in the present scenario, there aren't any except for what the bronze age civilization already has. In the region in question: cattle, pigs, goats, dogs, cats perhaps sheep according to wikipedia but crucially no horses; and those domesticated animals would probably still be much less docile than the modern variants we are used to. With regard to plants, the picture is not much better with much less variety and much lower yields. Which means, most of the population will be occupied with agricultural tasks at all times, except, of course, for those engaged in Hor-Medjedu's building projects. You may think that this labor could instead be dedicated to further technological progress, but Hor-Medjedu would not be very enthusiastic about this prospect. The picture is complemented with lack of everyday skills on the part of the time travellers from the present and by the potential presence of other antagonistic individuals, each with their own agenda and little patience for scientific experimentation by the protagonists for the purpose of recovering advanced technologies.
Still, provided that the time travellers survive their integration into bronze age society (Is that likely?), they should be able to give the ancient civilization a technologic boost. It should be unproblematic to write down a lot of theoretical knowledge, say, the basics of modern mathematics, to devise an education system that actually works, and to provide subsequent generations with a few guiding facts about what remains to discover scientifically and how that is supposed to fit together. Does 1k years sound reasonable as a time frame for everything else?
Note that there were a few questions that touched on some aspects of the present question: Here the question was, how long it would take to build a spaceship when starting with a single human without technology (with one answer estimating 60 million work hours). Here the question was discussed how to build and power a computer in the bronze age (but with modern technology available to the protagonist). Here the question was how to speed up technological development with a time machine (but as a planned effort and with many round-trips).