200 years to reach technology requirements, 700 years to reach your city count and 1000 years to reach the population count.
So you can reach your target faster if you have cloning technology.
Figuring out minimum time for technology
A space fairing colony starts from zero. The reason? A technologically advanced society has a huge support base. To build a laptop requires an industry of millions of people. So what societal level can 600 people sustain? Then we can figure out how long it took us to upskill from there.
Let's take a quick detour to the cities. Manhatten is 1.636 million people according to google. Rome was 450,000 at it's peak in 25BC. Apparently Baghdad got to around 1 million in 735AD, and with the advent of mechanical help, London and the UK surpassed our 1.6 million threshold in the 1800's. (Data from here).
So what technology did they have in the 1800's? And how fast can we boostrap to it?
A quick brainstorm tells me that our society should probably have:
- Horse and Cart
- Large cross-ocean sailing vessels
All right. What does it take to build a horse and card? A carpenter and some animal keeper. Great, that's easy. The animal keeper is easy. So long as you have a beast of the burden on your space ship and a human on your ship, and your planet doesn't kill them, you can have a horse and rider.
Carpentry is a little more involved. Typically you'll need steel tools to build a cart. This means you need:
- Coal miners
- Iron miners
- Someone who understands geology sufficient to find said resources
- Furnaces capable of reaching melting point of iron
The furnaces will require bricks and mortar, coal and so on.
Then there's the other two. Building a ship is "just" a case of more carpentry. I haven't been able to get good figures on how many people, but it seems that building a "big" sailing ship took at least a few years for the industry in the 18th century. In a Jules Verne novel (the mysterious island), four people started with nothing and built up to about 18th century tech including a small (10 tonne) boat in four years. They had pottery, basic metalworking and carpentry. I suspect he knew more about this sort of thing than the people of today, and so I can conclude that if four people could do it in four years, your 600 people can get to - and maintain this level of technology fairly promptly. Particularly with the help of their spacecraft.
Trains are harder. They require more advanced metalurgy. But it's mostly just a difference in knowledge than requiring anything physically more advanced. You simply require people who knows what a lathe is, and that steam can be used under pressure, and then you can build the industry to make a train.
I think that a 600 person crew can probably bootstrap fairly quickly - as per the Jules Verne example above. HOWEVER - people from today would not have the required skills to bootstrap an 18th century civilisation. But with education and training, they could.
So how do we divide up your 600-person crew? I would say:
- 300 farmers
- 100 blacksmiths/"engineers" (as in people who worked on steam engines)
- 100 miners/geologists
- 100 carpenters/woodsmen
Voila, within a few years of settling down, you likely have the required industries to create a 1.6 million person city and 18th century tech.
So how long from there? Well it took us two centuries to the current tech level, and it would probably take less due to passed on knowledge (eg math and physics can be taught for a couple generations even if the tech doesn't exist yet). So it comes down to population.
Figuring out the breeding times
With 600 people you're dead. There's a fun article here, and it states that you need around 10,000 for sufficient genertic diversity.
Barring issues with that (eg maybe they are genetically engineered people to not have issues with inbreeding) then it's a case of time. Pulling some numbers from wikipedia, the human population went from 2 billion in 1927 to 4 billion in 1974. That means it doubled in about 50 years. A quick sanity check says it took 40 years to go from 3 to 6 billion. Close enough.
For three 1.6 million population cities, we probably require another equal amount in farmers. So let's assume we need to get up to, say, 10 million people. This is doubling 600 a hair over 14 times. At 50 years per doubling it's a total of around 700 years.
However you specify 500 million people as a "good target". A little more math and out pops the number 20 generations, or 1000 years.