After thinking about for some time
It is hard to tell really. If you remove motivation, funds out of list of actual problems and leave human resources and knowledge then it hard to tell.
Reason why it is hard to tell is mainly the knowledge of that time. Until last century humans really didn't had the proper knowledge for all that - thus they didn't and could not.
It means, you have to inject some additional knowledge to make things happen, and there begins the question of how much of it. Even with the answer - a reasonable amount, so that we can, in a year or two, teach almost illiterate workers to do their part in manufacturing chain of processes, it still does not help that much to determine lowest possible year.
As an example - electricity isn't such a hard thing to make, understanding how it works, principles and all that it has very litle to do with production of finished machine.
Hydrodynamic bearings, sufficiently high precision parts of round or flat shape it not that hard. For flat ones, you remember sequence and practice it few days and bum - mastery of production of A grade flats. Basic theory of that - okay a week of demonstration on exagregated clay and wood samples will build some workable virtual models in brains of worker-master.
Static electricity isn't that hard, but can be useful. Tesla coil to convert static electricity to low voltage, also isn't that hard to make it even at bronse age.
So they may not necessarly get the ability to develop and improve technologies, but it does not mean they can't make parts and assembly them.
The problem is very similar, if not identical, to bootstraping problem, there is some difference, but it also about simple actions to get something from nothing and use it to help yourself to build few more of next steps at least.
Bronse-copper mirror can be good enough and it can be used to focus ligth out of flat sheet just by pulling flat sheet by the middle. (GREENPOWERSCIENCE youtube channel did some similar experiment, to test Archimedes story, but can't find the video atm, and Archimedes is 200 years before Roman empire)
So that time(200BC-200AD) can be a good candidate - they had humans, they had the system of organising people, they had quite sofisticated knowledge(people with that knowledge to teach workers), they had access to ores, stones, land resources etc. After RE fell, there is a gap of low quality time - but still does not mean impossible, just depends. I do believe that RE had a chance, but they were totaly clueless that they had it - maybe there would be one country and hyperdrive space ships today if someone pushed them in rigth direction - knowledge and skills of that time in rigth place.
Egypt also had some chances, they had less knowledge, but they had the way to organise people in production chains, results of that still tourist attraction, I mean, it still stands.
NB
A lot of things can be done by hands. And I do not mean Artisan stuff.
That's quite importand moment, hands are a tool, very sensitive one, and we had it from very beginning. One can cut a part on metal lathe, producing a lot of metal chips, but after that for some very preciese stuff finish can done by hands with or without power tools(not for every 6D CNC cut part, but for spececial and useful cases). And after that it at precision limit, more than that makes little sense (due termal expansion and all that) except some science stuff.
Pipets which can be used to suck in and hold a single cell, (cool eh?) can be made out of glass pipe with quite simple procedure - local heating and pulling, cooling, pulling, heating. It also precursor for high vacuum pump. Glass isn't that hard if one knows how.
So from very old age people were capable, but they didn't know what and why. Best old old time is animal domestication period, which probably far beyond the scope of your timeline, but just saying.
On other hand it needs to mention that things we may think like pretty mundane or ordinary can be quite challenging and require a lot of reading experementations if one does not know how exactly. Because indirectly there a plenty of science backed in - as an example kf such items I can name a crucible (for iron melting).
So simplicity can be very deceiving.
Roman empire, anything during and after Renaissance can be a good time.
But again, what use the energy has, because by itself it does not give advantage. Making growing crops be more efficient, and introduce some simple but scientific selection process for plants, and pest control, and irrigation - can be a more potent change, just because you have more people who are free to do other things, even if it is no more different than it is typical for the time.
I would say since 200BC people were capable to do your sun tower.
Flr reasons I mention earlier.
However, electricity as means to transfer energy is not necessarly the best, until things progress a lot.
In usa pneumatic hand tools are(were) quite popular, and such stuff jndeed has certain advantages - simplicity of construction, weigth, distribytion of air(pipes) is lmost as flexible as electricity.
Hydraulics, just water, also quite potent solution for main line - your 10 MW source (which is just insane amount if power by that times)
As storage - some elevated pond, 50-100 meters up, can be decent accomulator, and that is about 10 cubis meter of water per second, few 10 inch pipes will do. Main pipelines not neccessarly have to be metal, basalt and diabase can be cast as well, and it is on pair with cast iron.
And water flow(trough pipe) can be converted in all sorts of energies.
For bearings - use hydrodynamic or static bearings, made with high precision out of stone - no special materials required.
So time really depends on your capacity to build tech chain for what you point as advantage. Selection of what will be advantage also depends on you.
If no new knowledge, except the concept, and probably simlpiest steam engine aka tesla turbine and bearings for it - then it is time when you have decent pipes and mirrors. Early 1800 may do probably - but the thing will have hard time to compete with that time burning for energy ways.
So you propabple have to choose before steam engine time. Basically making that named steam engine, it being the key piece of tech. Which not so much different from refferencing how things were rolling after that, and without electricity part it can naturally happen since 1300AD
So really there are no easy ways to point at some time. And only your handwaving will keep the tower to be replaced by a typical boiler at some less speciefic and more convinient place.
P.S.
Similarity with bootstraping drove me for most of the answer in direction to "make it happen", rather "sufficient chances of such thing happen naturally", but your time options do not seem to be consistent whith that as well, so maybe it is okay.