I dont know about BA but the only rules there are that aircrafts have to be at 1000ft AAL within 6.5 km from start of takeoff run, minimum 4% climb gradient to 4000ft, less than 94 dBA at the noise monitors and to follow the noise preferential routes. (UK AIP EGLL AD 2.21)
To be honest I find it peculiar that such an airport simply doesn't simply have a bit clearer rules such as NADP 1/2 or some versions thereof