A few rumours to knock on the head - the selection process does not favour children of current BA staff - not only do HR not particularly like them, but attitude has got in the way of a fair few in the past. Selection is equitable and meritocratic.
There is no scheme on the side by any means - there is only selection after recommendation from flight training schools, which though ad hoc, was above board and not subject to favouratism.
Furthermore, there is no budget for it at all. Whatsoever.