The world as we know it as ended, and the vast majority of humanity is gone. There has been a period of turmoil after the apocalypse, but order has now been restored by a ruler. You are a member of a surviving community of a few thousand people, in a largely peaceful society. Given the appropriate guidance, what do you think is the quickest that such a society could progress from rudimentary means to a technologically competent civilization with, say, electricity, radio, and engines?
Answer is: Which civilization are you talking about?
Okay, civilization has fallen and order has been restored by "a ruler"... I'm going to take a wild guess and figure that no, a Constitutional Republic has not risen up to replace the one we currently have in the aftermath of whatever it was that took us down to begin with. These United States grew out of the Enlightenment, thought up by enlightenment thinkers - some pretty heavy intellectual hitters who based their reasoning on the writings of even earlier heavyweights.
Since guys like John Adams and Thomas Jefferson aren't thick as pebbles on the ground, I'm guessing that something more primitive than a Constitutional Republic has risen up. Probably feudal. One strong man, set up as "king" or grand high potentate or whatever he chooses to call himself. Under him will be trusted lieutenants controlling smaller areas and under them henchmen of various ranks, busily henching away. Those who are not busy henching might make up a technical class - those who know things, can make things, those who can teach others, repair things. Most valued will be chemists, engineers of varying flavors, doctors, machinists, etc. Got to have some sort of an economy to fund all this - unless the Head Honcho is a psychotic tyrant and gets what he wants through fear and intimidation - so there's gonna be a merchant class. Farmers and ranchers will probably make up their own class. Below these guys will be laborers, various levels of peons, etc.
There's all sorts of variables to take into account. How many people of the technical class do you have available? What's their skill level? How many peons and laborers do you have to build stuff? What access do you have to old tech - books, manuals, etc - ? You're going to have to have some type of schooling, since even the schmoe workers will have to be able to read, soldiers will have to read manuals, do sums, etc. Plus, you're going to want to be able to pass information to the next generation, so you're at least going to need a printing press and someone to teach the basics...
Man, I honestly don't know... not that I wouldn't at least try to give us a leg up. The alternative - rediscovering everything from zero - will take ages. If we had one guy with vision, motivating everyone and life didn't suck so bad that folks up and left or started a revolution? At least a couple generations. Call it 100 years.
You can't go from chucking pointy sticks at your dinner to hydroelectric power plants in one leap...