First consider what fallout actually is: basically any material blown into the upper atmosphere during a nuclear blast -- which eventually falls out of the sky. Essentially fallout is a bunch of ejecta flying outwards from the epicentre of the explosion. (In fact, fallout in space is kind of a contradiction, but let's ignore that!)
So, what happens to a bunch of ejecta flying outwards from an explosion in space? The answer to that becomes obvious if you simply consider each particle separately and dust off your old celestial mechanics textbook. Each particle will have its own orbital trajectory, with most of them flying backwards in a lower orbit than the spacecraft (this is how all rocket motors generate thrust, through momentum transfer). Some will also be tossed outwards laterally into more eccentric or inclined orbits, but it will not simply remain suspended in the vehicle's wake.
Next, solar wind will most certainly have an effect. However, it is difficult to say whether or not that effect is significant enough that you could easily discern it from the effects of gravity.
Finally, it depends on how close you are to the nearest celestial bodies, but the fallout will generally enter into orbits around whatever central body the vehicle is traveling around. So if you were traveling in interplanetary space the fallout would orbit the Sun, whereas if you were traveling in interstellar space then the fallout would orbit the Milky Way. Some may fall back to the solar system (hey, maybe fallout in space isn't quite contradictory!) but yes, it would also fall behind the solar system.