I guess the question needs to be asked in two parts.
1) How would they react to the modern world if they were time warped here - Like your one story Tim about about them being whisked away moments before death in the Ipatiev basement.
2) How would they react to the modern world if they had grown into it some - Like imagining they'd all been exiled instead of killed, and then imagining that Alexei could have somehow survived long enough to be treated with the first major hemophilia treatment innovations that appeared decades later. I guess you sort of touched on this in your "Bid Time" story as well Tim, showing OTMA in their 30s and N&A as grandparents, yes?
Obviously none of them would still be alive today. But I could have imagined the daughters, at least, living into their 90s and perhaps even their younger brother dying only a couple of decades earlier. But I'll keep the focus on option #1.
Something tells me that unless desperate for income they'd never accept being parted from each other. We could sell their priceless jewels for huge sums of money that they could surely live off of, for a while at least. Perpetuating the lie to the world would be tricky. What with social security registries, insurance and government regulations and all.
Nicholas could work his remaining years on a farm registered under my name. I don't feel like Alexandra would have a hard time either adapting to the peace and quiet of the countryside...hopefully she'd be able to make a few friends. The girls would surely want their independence at some point, but if captivity over the course of sixteenth months taught us anything about their personalities it's that they seemed perfectly happy around each other, and seemed to have little problem living in a half decent home and removed from the outside world. They could do with less. I imagine them living similar to how a Mennonite Family does now (Amish might be a bit extreme for their tastes). Perhaps they could eventually be brought into that little community, marry, and start families of their own. My wife's childhood home that her mother still occupies (and lives next her brother and sister in law) has an Amish family living across the street and two Mennonite families on either side. It wouldn't seem so far fetched...