In my opinion, I think the world would have a been rebuilt a bit more than it is in fallout 3. Even DC being the main target for nuclear bombs, there are still a lot of other regions that would have not been bombed and would be able to aid in the clean up and recivilization of the area.
While the half-life of fallout from nuclear bombs isn't measured in hundreds of years, you still have to remember just how many bombs were in use. The amount of radioactive matter that has to dissipate before the world becomes safe is mind-bogglingly huge. This fallout isn't just going to be at the blast sites either. It's going to be caught with the wind and be moved around all over the planet, particularly when the entire world is being bombed at the same time. What regions would've been safe from that fallout?
And another question, how exactly do you clean up after a nuclear apocalypse? It takes eons for things to not contain lethal doses of radiation and by then almost all the vehicles have rusted to death and there's largely no fuel for them anyway. How do you clean away that much rubble? What do you do with all the rubble? Where do you take it? And why bother?
200 years is a long time. Where were we in 1810? Riding around on horseback. Compound fracture? Cut the limb off. Muskets as weapons. Life expectancy of around 40. You get my point.
200 years sounds like a long time but you've got to remember that there's also been a lot of people around, working hard to contribute. There's nearly 7 billion people living on the planet at this point. At the time of FO3 there's probably at most one or two million people left alive in North America, probably much less than that. New California Republic is the largest town in America with a massive... 3000 inhabitants.
There were some 120 vaults with at most a thousand people in each, probably much fewer than that. How many could possibly live outside the vaults? Not only would those people need to survive the blasts (and being in a metro or mountain cave or merely a solidly constructed concrete basemant might with extreme luck provide that opportunity), but they'd also need food supplies and indoor shelter for at least a few years, or the fallout would do what the blasts didn't.
I would give it about 50-75 years.
Are they completely done cleaning up after Katrina in New Orleans? It's been 5 years now, and that's in essentially the wealthiest and most powerful country in the world, and without a massive decimation of the population. Increase the damage by a factor of at least a thousand and reduce the population to one percent. You still think it wouldn't take more than 50 years to clean up everything?