Since they live at hundreds of times normal human speeds any nanite starship would by definition have to be a generation ship or a sleeper ship, totally self-contained with no chance of returning to the culture that produced it. Even at warp speed the shortest interstellar journey would last many generations.
Unless, of course, they combined space and time travel. Travel out to the stars at a (to a nanite) crawl, with the passengers and crew in the nanite equivalent of suspended animation, then travel back in time to a few seconds after they left their homeworld (unless they really want to screw up causality they won't travel back to an earlier time, in case someone decided to travel back to the home world and arrive before he left). If you want to interact with something in the macroscopic universe (such as Jean Luc Picard) you use a robot probe which from the nanite viewpoint is heading into the extremely distant future. Tell it to say or do something, observe the results, then return to your time. If you don't like the results program it a little differently and send it to a slightly earlier time, so that the timeline it creates will overwrite the original. Repeat until you like the look of the future.
It's a VERY strange viewpoint, but it seems implicit in the nature of nanites and the Star Trek universe. If you don't like the way the future is turning out, change it before it happens.
I know that this would NOT be very popular with Hawking etc., and that even the nanites would have to be careful not to screw up their own causality, but essentially they'd simply be tailoring the legs of the Trousers of Time (©T. Pratchett) before going down them.
Am I missing something?