This falls into my "how many angels can dance on the head of a pin? Answer: As many as wanting" category. The numbers you're looking for are small enough that the world's manufacturing processes can easily absorb construction. But there are a couple of assumptions.
Assumption #1: Earth is completely united.
Meaning the entire world's scientific and manufacturing capacity is at your disposal.
Assumption #2: The existing military-industrial complexes and corporate-industrial complexes were not substantially destroyed.
Kinda ditto Assumption #1.
Assumption #3: You didn't have these ships to begin with. You're designing, as it were, "from scratch." This would be the equivalent of designing and producing a new ship from concept to the broken bottle of champagne.
These assumptions may not be at all the reality of your story, but they produce a worst-case-estimate that allows you to scale back the construction time to suit your story.
We're also going to need to change your design a little bit. There are some truths to space flight under the conditions you're describing that will limit us.
- We can use real-tech nuclear reactors as found on nuclear submarines, but this means your small ships will NOT be the size of mobile homes. They'll be the size (at best) of nuclear submarines. This is actually a reasonably realistic conclusion. And here's where your life begins to really feel pain.
Taking 14 years to design and build and costing around £1bn ($1.6bn) , the British nuclear submarine Astute is one of the most technologically advanced and controversial machines in the world. Weighing in at 7,000 tonnes, measuring 100 metres (330ft) long and boasting three decks, how do you put such a craft together? (Source)
Nevertheless... rail guns for torpedoes, ion engines for propellers, use disposable fuel tanks and external engines to get the sub into orbit (not impossible, but I leave it as an exercise for the user to figure out if this is practically possible) and you have your small ship.
- Gratefully, government satellites take an average of only 7.5 years from design to construction which means the space-tech part of this fits within the design limitations of the
submarine space ship.
So... 14 years for one small space ship.
- I couldn't find an easy spec for the number of submarine-capable shipyards in the world, but there are 26 countries with shipyards. I doubt there are 26 sub-capable shipyards, but let's run with the number. 210/26 * 14 = 113 years.
113 years to build 210 small ships
And it's worth noting... there are approximately the same number of astronaut-bearing space agencies in the world, so it's plausible that you could build 26 ships at a time.
- Your big ships are 4X the size of your small ships. It's a guess as to whether or not efficiency scales. It often doesn't when it comes to construction, but let's say it does. So, 2X the time. You're using the same shipyard resources. 90/26 * 14 * 2 = 97 years
97 Years to build your big ships
- That sub I mentioned is only 100 meters long. Your big platforms are 500 meters. So, 5X the size. The smaller platforms are 30 meters or 0.3X the size. You didn't mention quantities... but they would be a hair easier as they wouldn't need substantial propulsion. GUESS: say 1.5X for the big platforms and 0.15X for the small platforms, meaning each group of 26 big platforms add 26 years to the total and each group of 26 small platforms add about 2 years.
Whether or not it takes less time depends on just how far into the future your technology goes. It takes time to build big, complicated things.
Whether or not it takes more time depends on just how many of the construction resources (from primary manufacturing down to supporting resources, like raw mining) were destroyed. If the aliens destroyed 90% of the capacity you need to 10X the time.
I don't have any idea how quickly new shipyards could be brought online. They're BIG. But the odds are that in a world-wide panic you might cut your ship building time in half by building shipyards as fast as possible. NOTE that this would be DEVASTATING to national economies. In fact, just cranking out the ships and satellites will very likely be devastating. I don't know how fast (for example) the U.S. retooled for defense production during WWII, but compared to the total length of the war, it was a non-trivial amount of time and had massive impacts on both the economy and on society — for both good and ill. Impacts that we are still bearing today almost a century later.
One point to remember! That 14 year number is concept-to-water. Theoretically that's the time for the first ship. It gets faster with each succeeding ship as kinks in the process get ironed out, efficiencies fall into place, etc. On the other hand, there's also the very high possibility that the Mark I ship gets replaced by the Mark II ship... etc... meaning that it's not all a bed of roses. Remember, this is your worst-case number. You modify it to your needs.
But, all things being equal: about 200 years.