Travel to London would have been relatively safe, it wasn't until a few years later that regions of the south east were in open rebellion.

Travelling north to York or beyond would have been dangerous if by land (ship was popular). The north of England maintained a guerrilla conflict with the Norman invasion in an attempt to reassert their independance from the south and 1070 was the height of the campaign of the slash and burn policy that William inflicted on the area. He burnt and destroyed the area leading to mass starvation and death in an attempt to weaken the rebels who he had been unable to bring to battle. The doomsday book (almost 20 years later) still records over 50% of the area as "wasted" and only a quarter of the population still alive. A Norman from the south would not have been welcomed by the Anglo-Scandinavian population of the north, as well as being vulnerable in a "lawless area" that was being subject to Scottish and Scandinavian raids.

You would also travelling before the northern lords were replaced, so protection from the upper classes could be limited, unless you were travelling along with the Kings army.

William Kapelle - The Norman Conquest of the North