The UK is not the only country with advisory hotlines for air carriers. To my knowledge, at least the US, Canada and Germany have a similar formal system. Those numbers are confidential and if one were able to obtain the number, nobody would speak with you anyway. It is the most efficient way to handle the issue and ultimately all that matters is that the relevant immigration authority authorizes the air carrier to board the individual in the knowledge that it will not be penalized for doing so. Other than satisfying OP's view that BA and UK immigration are a pack of liars, it does not inform or help the situation.