Hi Julian,
What exactly do you mean they are in the same coordinate system and geoid? Any Bing service should be using WGS_1984_Web_Mercator_Auxiliary_Sphere on GCS_WGS_1984 datum. Are you saying that you projected your OS data to that coordinate system? What was it originally if you did project. The OS data I'm familiar with uses OSGB36 datum which can vary from WGS84 by as much as 120m. The shift you describe could be because the appropriate geographic transformation was not set at the time you projected the data, if you did...
Best,
Eric