On considering the model of the American Colonization of Europe, don’t forget that that method included a continual influx, over hundreds of years of people, and materiel from the parent society. (1609-1789 as colonies under rule of England, and later as a destination of millions more even into the present.)
Remember as well that the Roanoke colony perished without a trace, and the Jamestown and Plymouth colonies would have as well without support from later contacts with Europe. If we are going to suppose annual shuttles from home, this problem becomes trite in the extreme. Without any further commerce it is a very difficult problem.
I think you have to ensure the continuity of culture first. Without it, technology will erode, either sooner, or later during crisis periods, whatever their source. I also think trying to exactly duplicate the technological model extant in our world would be ill advised for many reasons. We could do better, and it might be a lot faster if we were to do so. Our technology is based in very large measure on the centralization of population in cities. That is an inheritance of a cultural pattern that predates the industrial revolution.
Breaking that mold would make low impact, more easily reproducible technologies sufficient to provide high standards of living for multiple populations that can grow gradually, but with the benefit of decentralization to minimize the inevitable losses from disasters. It would reduce the transmission rates for epidemics of disease, as well. Planning prior to departure is critical in this matter.
Using the Mesopotamia without hominids world suggested provides us with a lot of specifically useful information which is (although absurdly unrealistic) of vast importance in such planning. The soft metal equipment, and low temperature smelting technology can be originally imported to the exact locale of the most ancient mining sites known in the region. If early bronze age miners could get early bronze age bronze out of this, certainly our intrepid colonists could be drawing wire, and making brass clockworks in pretty short order. Steel is nice, but there are a lot of very useful things which can be built without it. Glass production can be in place in short order as well, using only the same resources available to the artisans of 5000 years ago. We know where the oil is, by the way, and we know what can be done with every single tiny fraction of it.
Since we don’t need to search for eikhorn wheat, or such things, we can move into the Nile delta in just a very short time. We can have good agriculture abundance without plows or draft animals, if we practice the best of our horticultural technologies on multiple site small scale farms connected by seasonal river traffic. The abundance of game can be managed without the drive to compete with neighbors, and sustained for thousands of years. Keep in mind that a thousand miles north of here there are Aurochs roaming the primeval Asian forest. Given that good sailing ships are pretty low tech, especially once we get cloth making and optical equipment set up, we can disperse a population of a thousand or so at a time to new locations along the Mediterranean and other waterways in a few dozen years.
Without waiting for steel production, which is a very high cost endeavor, we can still have copper, silver, and electrical equipment of high levels of reliability within a generation. Steam power is great, but it isn’t needed for a long time, and we can leapfrog over to better methods before we invest so much in the mining of coal and iron.
Brass faced wooden gears can run mechanical mills powered by wind, water, and solar power to provide clean cheap mechanical energy in a distributed power system not reliant on large grids, or huge utility companies. As our manufacturing output increases, those same facilities can produce electricity. The existence of that power source will make electronics a far better investment than automobiles. The only real need for intense power application such as the automobile would be for railroads, which will require very large populations to build and maintain.
I think three hundred years, mostly because you really do need to get a very large population spread entirely around the Med. But by then I would expect to have flight, rail, and powered ships supporting a varied resource base, and a culture beginning to examine the desirability of recreating nuclear technologies. I would hope that the society created would have decided to let some portion of the Earth, perhaps the Americas, to go untouched. Perhaps a culture force to live with a common need to maintain its every ability and talent would prize its individual members enough to make their nurture its first priority.
Or, we could all go back to Mesopotamia and burn that great library like we did last time.
