Dicky, do you think the census could have taken longer than one year, the year in which it is thought to have happened (6 AD)? If so, perhaps it ended at that time but Joseph could have made the journey to register 6 years earlier?
It says that the census was taken while Quirinius was governor of Judea. That firmly dates it to 6CE or later. More to the point, the decree allegedly emanates from Augustus and is a census of the whole World (Roman empire presumably). There's no evidence that this census ever took place.
More problems for the census:
Galilee was a client kingdom not directly under control of Rome.
If Jews had been allowed to travel from a client kingdom (Galilee) to a Roman province (Judea) to register, it would have rendered the census useless. Not only that but the economic disruption would have been a disaster.
The census is merely Luke's plot device to get Jesus to be both born in Bethlehem and from Nazareth.