TLDR 100-500 years.
bronze age - hm, I was absolutely carried away from that one, yeah, oh well, hm, but still there is some truth in my answer, lol
- For motivation, the answer is "motivated enough". I'm not meaning to sound flippant, but the idea is that I want this end result, and I'll tweak almost any factor that needs tweaking to get there. ... (op)
Okay then you have an Amish-like society, an internally coherent group of people lead by Moses and nourished by the black monolith and then it is equivalent to a typical question of bootstrapping a colony on another habitable planet. There are questions of that kind here on wb and elsewhere - helping time travelers to survive, rise technologies in societies, space colonies establishing, etc. Quite scattered material and usually not that great, but suggest to look it up, for some inspirations and information.
A note on growth rate.
There was growth rate concern, 2 or 4 % may be too big, etc, I have to point out that, let's say 4%, is absolutely not a limit for growth rates of humans. 6 children per family, it is about 12 years in pregnancy cycles, is not something astonishing in old days(it is more a perception of mine, so as some facts, including actual growth rates and numbers for children mortality rates back then, for some places)
So 6 children per family on average, means a growth rate of 5.6%. So 2 or 4% aren't high, especially considering that simple hygiene procedures, simple disinfection means like C2H5OH, understanding of germ theory can help drastically to slash that child mortality rate, women's mortality rate, and overall. So simple procedures which do not require equipment and are just a result of education and creation of habits(like do not flush "dirty" water on the heads of pedestrians, keep rodent count low, burn or compost trash, quarantine people, trace contacts back once epidemy, etc) can lead to significant improvement of the epidemiological situation to the point being able to reduce risks of epidemics and even fight it to some degree(vaccines are better of course, but ...). Simple means can be quite effective.
Agriculture, farming(rotation of land, compatibility of plants predecessor cultures, or permacultures strategies), and selection work for plants, irrigations, etc - it also can be helpful knowledge that not necessarily requires some dedicated tools or chemistry but it more about workflow and what to do and what not to do. This knowledge helps worry less about famines and such things.
There is plenty of knowledge of that sort which is helpful to increase productivity and reduce risks as a result of rational planning and knowing what to do. And this or another way, in different shapes and forms all that had some occurrences in the past - like domestication of animals was selective breeding, etc. And as a whole system, it all can have a compound effect - having it in one place at the same time is handwavium, but elements of it are not.
In that sense, 4 percent can be a quite relax number and not stretching anything.
Technological advancement is the first goal from the very beginning. The reason for that is that it reduces efforts one spends on food production and be able to spend the rest of the time on other activities.
There are some bottleneck factors, like location and proximity for ores materials, fertile land, river energy. A good location could be a foot of a mountain with a relatively fast creek, next to a fertile flat thing next to a big enough river, with ore veins and dried up the ex-ocean bottom with Manganese nodule and a big forest nearby. That's one of the options.
The problem of bootstrapping, or it possible to say its advantage when one knows the path of development 100 steps ahead is that it does not have to repeat the history, it can be done differently, cutting corners and much faster with much less effort as one does not have to do 95-99% percent of the R&D process, no dead-end routes, planning ahead is possible.
The chosen way affects the speed of development, not in the last place because different paths have a different set of conditions - restriction, bottlenecks, opportunities.
A hundred people can't do much in terms of technologies, they do not have enough man-hours and they have to busy themselves with food-collecting, but they can do some which will clearly distinguish them from just hunter-gatherer communities.
But you have to have some plan, a route for them, and it is not that easy to make a reasonable one. As an example making them farmers from the start maybe not be such a great idea, as fishing can be a gold mine in terms of energy return on energy invested(EROEI).
And the path of development has to be optimized in that regard. Not efficiency of production which we care about today, not a price which is convoluted mixing pot result of a lot of things including those which have nothing to do with EROEI. This very EROEI is the main optimization parameter.
- to give an extreme unrealistic example, if to make a first electricity generator you need 1kg of iron and you can make it in two ways spend 20x effort to get 10kg of iron, or burn a forest with ease and get just 1kg - most likely you burn a forest - a path of least resistance, laziness is a driver of progress - pretty much humanity lived with these mottos since the very beginning, and it turns out it can be a rational choice, despite the extinction of species due hunting and creation of some problems in the future.
With planning, they do not necessarily need that, but planning itself is created in the spirit of that.
- omg, so much text already, let's move to the business, the topic has no end, very exciting...
stages of development, sizes
The introduction was not sufficiently comprehensive, but there are limits to how much fits in one post. The main point was development route can be optimized in terms of efforts required to achieve certain technological progress, and before knowledge which eliminates dead ends and R&D efforts is a great boost. So they are (or can be) constantly in a state of knowing what can be done with what they have exceeding their real capacities to implement things, what they actually can do. Pretty much never was a thing in our human history, and when such thing raised its head a little - we call it industrial revolution since then.
So a situation is limited by the number of people present for work doing, which is a function of the growth of population. What we may have problems with is to be able to set points on the plot which correspond to certain, known to us, technological levels.
A hundred people can advance in technological means quite a lot actually, despite it being just a typical tribe size or a village it would be wrong to make parallels with actual villages in old days, and a more correct way would be what can 10 guys of modern time do, assuming they are provided with the knowledge of what to do, and maybe some dream training by matrix(extreme case) black monolith - meaning they know what to do and skilled enough at doing it(not necessarily a stellar job, it may take a few times more than for a professional but they are motivated and dedicated enough).
So it 10 of time travelers(the rest are working on food child caring education material delivery etc)
- borrowing energy from external sources and replacing human physical labor was one of the fundamental achievements, which made farming better as it more energy expensive than hunting(if I recall it correctly, and didn't screw my numbers back then)
So domestication is one of the things which hard to speed up, effects require 50-100 years, but can they borrow river and creek energy for mechanical work - absolutely. The tools required for that can be just a stone(sort of a stone ax) or even just hands and laying trees.
Basalt or diabase can be cast similarly to iron or how other metals are cast. So as with mechanical work stone can be worked to all sorts of things. Combined with wood it can be a replacement of a lot of metal things in sense of machinery and mechanical contraptions.
- can accept a challenge to make a plan for 10 modern humans how to get from stone and wood and river to electricity and metals without prospecting for metal ores and making mine shafts and such.
Prospecting for materials is one of bootstrapping bottlenecks, but there are ways around it, so as places rich with gold also provide metals as well - so you can pan as for gold so as for iron oxides(and other metals in that black sand) so getting a little bit with a minimum of tools, isn't something unrealistic.
What is the limit for those 10 guys, it depends, electricity is achievable, practical uses(however limited) achievable, steam engines, cryogenic setups, acids, rectifications of all kinds, bearings, abrasive materials, metals, etc.
A lot can be done, but there is another problem how many things they can support running and operational at the same, with servicing and maintenance works. They can solve complex problems step by step with different means with a goal to get to some useful plateau which they are capable to support and which is more or less self-sufficient.
The level of technologies defines not only what is possible, but also the percentage of people busy with food production, less is the number more time they have to build cities.
Imho, it may look optimistic or not, but you may guess I have some reasons to be optimistic, lol
Limits improved 1400 A.C. level of tech maybe improved early steam era, water wheels or turbines(simple profiles, not necessarily Pelton turbine, but close to that) ones are not impossible and in terms of mechanical energy 10 - 100 kW are not impossible (if there is that mountain creek of sufficient power, quite a good location for initial energy generation)
Even 10 kW is a lot, for grinding of something, cutting stone blocks, drilling, etc - it can and will outperform the whole village if they would do that manually. Not that they need stone bricks at this point, just saying.
1000 people (50 people in charge of technologies, something like 20 percent of who are not kids or elderly)
It is still a village. They can expand on the variety of things they have or do, can have refrigerators may be in form of improved under earth ice storages keeping things spoil free with no gaps in time, all year round.
Overall they may have essential technologies on the level of the 1900s, not all, of them, but combustion engines are reality. So mobile energy conversion units for all sorts of works are a possibility.
Metall working shop is a thing, production of metals in sufficient quantity and parts from it.
No cities as of yet, a city requires brute force, but it starts.
10000 people (1500 in charge of technologies, 50% of workforce which not busy with food production, and another 1500 season workers another 50%)
At this point variety of implemented technologies can be quite decent, with exception of a lot of production and products which is not required yet (like radio, telegraph, airplanes, personal cars, a lot of chemistry, etc, etc) those guys can support a subset of 1930-40 technologies and it is a sizable workforce which is capable to build and support a city for themselves, at more or less modern standards as appearance and some functions, maybe without some bells and whistles, but with tap water, central heating, cogeneration of energy(thermal electricity). The assortment of goods may be poor or definitely not stellar, but in a sense of survivability, and medical necessities including vaccination can be advanced enough. So with sufficient laser focus on necessities, they probably can draw technologies from 1930 to 1970s. No computers, no tv, no microwaves - those are not necessities. Electric motors and their easy derivatives like blenders no problem, combustion engines, and shiny shovels.
So if you slash what can be found in some nowhere place with 10k people in it, and upgrade some things if you are looking at USA reality of such places - then it something you could expect from 10k people. If they work like one solid company, be focused, and have energy.
Energy is important, sticking to instructions step by step what needs to be done in a workplace begins to be a necessity, rare elements begin to be essential for further technological development.
Energy requirements at that point a few MW's will cover their needs, and operational activities. Biodiesel may be a thing for them, as it is more convenient than splitting activity for oil prospecting, and has some advantages over coal.
Wood for power at those consumption rates about 50 tones of air-dried wood per 24 hours, with 25% efficiency for electricity production, rest for heating drying and other low potential heat uses (keeping warm water in fish farms as an example, greenhouses, whatever)
For that it needs about 10000 ha or 100 km2 of forest(it grows that much(50t) of wood a day, kinda), so even if it is just wood - deforestation like it happened in Europe and which moved them to more coal use, is not a thing for that settlement. It is another example of the significant differences and problems which historical analogies can bring in a bootstrapping situation.
Their demands for materials still can be covered locally, and extracting 10% iron-rich ore, which we find not economical as of today(50-60 is today's norm, 80% was like 100-150 years ago norm) can be a better option for them than splitting and invest in roads and long-distance transportation.
They also still have enough mobility and if there is such a necessity or opportunity, it may be beneficial to move as a group to a new place - sure it takes time and effort, and it abandons a lot of earlier investments, but if someplace else is a better choice for a current situation they are capable to do so, same as they were capable to do so before as well, for a next stage it is a much harder choice.
100'000, 10% of population is food production(farming), the rest is technology(20k), building(low skill labor), kids(low skill labor) and elderly(easy labor, edication, managment, processes overseeing).
Pretty much expansion of what they could do at the previous stage, the city they can build and maintain should probably satisfy most of what one could expect from a city. Roads and satellite cities for farming. Trucks for cargo, centralized storage facilities for all sorts of things. Public transportation available(Trolleybus - there is some simplicity in the thing, even if it is less flexible, and has certain limitations), as personal transportation a bike probably still a thing, but it depends on how much energy they have.
for goal stated in q it does not need that much, but as an honorable mention
1'000'000, Musk-time - sufficient to support modern technologies self-sufficient Mars colony
40'000'000 bare minimum for the modern technological country, sorry Norway.
1B - capable to support the production and development of all modern technologies without a pinch of black magic. The number can probably be a few times lower without the necessity to develop and research. The absence of competition as a driving force of development can probably slash the number another few times.
if conditions are a-okay.
If there are not many factors that may cause some significant setback and limitations, which does not mean the absence of hardships, then somewhere in between 10'000 to 100'000 you will get what you want.
With 4% it is 120 to 180 years.
With 2% it is 230 to 350 years
For 1%-guys, it is 460 to 700 years.
For 5%-rabbits, it is 94 and 140 years.
So 500 years seems to be not unreasonable by any standard, and more or less it is what happened in history, I mean industrial revolution started about 260 years ago. They had different starting conditions, they had more people and less knowledge but the speed of implementing knowledge(once they produced it) was quite fast.
So from a certain point of view, a few hundred years in your situation is not necessarily an unreasonable expectation.
Lack of numbers is a problem but at the same time a blessing because they require fewer resources to implement things to be available for everyone. This means they can cut some costs and afford more inefficiencies, working with smaller ore mines or sticking to other means of resource extraction, like electrolysis of basalt which surprisingly contains plenty of stuff Basalt::Geochemistry and despite processes are more energy-intensive(not necessarily by much, but 2x) but at the same time it slashing costs on a lot of things and just saying roads and transportation does not describe situation deep enough. And one still can afford to be a little bit wasteful and still sustain this activity on local resources.
Clay bricks may require finding proper clay deposits, but the same basalt can be cast in bricks - again maybe a little bit wasteful, but hey it won't be like in Rome when buildings were crumbling cuz someone slashed costs on baking the bricks, those basalt bricks will last for ages, and if one casts them as lego blocks like wavy surface they may not even need mortar to make a building, meaning those bricks can be a reusable material.
So a little bit of creativity and unorthodox(from a modern or historical point of view) technological thinking can bring decent results.
- seems I didn't address it well enough, carried away by technologies and bootstrapping
here I have a little bit of problem in terms what do you mean by walled city and hope you may draw some estimations from technological capacities.
But let's say a middle-aged city with a wall - there is a choice to make - one can spend a hundred years to make one from nearby rocks, castles were done in that ways, or invest some time to make tools which will make the thing easier and less dependant on proper stones laying around.
Things depend on choices and the environment. if it is an outpost-like situation, when you build a wall against animals or other stuff - then things do not allow that much freedom in terms of slacking off and build tools, you start building your wall straight away.
a thousand people can build a walled city for their size, even a hundred may and in some places did.
At 100k city a wall loses its meaning if it is against an intelligent enemy.
so depends on the purpose and environment but 100 to 100k they are all capable to build this or another version of walled settlements.
when we talk about the fastest way to do so then it needs more specifics but it is not necessarily possible to say that technology is always a faster way to do so, or more technology is a faster way. it more like a balance of what makes sense and how big is the task, what the end goal is, and there is no general answer. here is plenty of examples where simpler technologies do provide a satisfactory result with much lesser efforts than more advanced one which does the same.
10k is a good population number it allows plenty of things to be done, in different ways - old old, old, modern, or bootstrapping.