Thanks, Dylside--I'll have to think about all that.
Nuclear war: outside may or may not be a bad idea. I gather that MOST of the radiation would be of a very-temporary kind, so once you got past the initial explosions and stayed out of the dust for 72 hours or so, it wouldn't be radiation issues you were dealing with, except in the immediate area of an explosion (where you'd have a concentration of the radioactive elements with massive half-lives).
Bike idea: might work best as part of an overall plan. Bike advantages you've noted: extreme mobility, high gas mileage. Downsides: can't carry much gear; hard to carry more than 1 passenger, and must be pretty able-bodied. You mention the existence of a wife. One thing I think about is, what if one of us were injured? It's easier to transport someone injured in a car, I'd imagine.
Kids--when they come along--are a whole new level of complexity. I guess you could strap even a baby onto your back, papoose-style--but that's a limited number of kids.
As I think this through--thinking of my own situation, with a pregnant wife and 4 kids ranging from 1 to 7 years old--the bike would be of interest for mobility, but might work best in conjunction with a higher-capacity vehicle. I could see using a 4 wheel drive SUV for, say, an initial evacuation, and then the bike for forays out after food, etc. Could also imagine ferrying one passenger at a time to some bugout spot, if roads were impassible.