The BBFC is a bit of a mixture.
For the cinema, the statutory power is held by local authorities, who accept the BBFC rating as guidance, but can set a different rating, or permit to be shown a film refused a rating by the BBFC. Though the Extreme Porn business has given some special status to a BBFC rating.
For videos, the BBFC gives a rating which is backed by law, and this is the starting point for rating of games. At least initially, it was the cut-scenes in the game which could get the BBFC involved.
Personally, I'm not sure the BBFC or ELSPA have really sorted out their thinking: participation by the player makes a game different from a film, and the great experience of the BBFC may be misleading.
Then, at some point, there will be a Wii attachment and a related game with an explicitly sexual purpose. I hope to be able to watch the argument from a safe distance: another planet, perhaps.