I am not 100% sure that the rule that Fraser quoted is entirely correct since it mentions one world lounges. The published one world rule is

"Customers flying First or Business Class have access to the equivalent class of lounge regardless of their frequent flyer status. (First Class passengers may be accompanied by one guest travelling on a oneworld sold/operated flight).

Customers flying in Economy class on the domestic or intra-European segment to connect to or from a oneworld longhaul First or Business Class flight can also use the lounge applicable to the class in which they are travelling the longhaul segment."

Unless BA has a specific rule that allows access beyond this when travelling BA then lounge access when connecting to economy class only applies when travelling longhaul business to european economy

Dave