The fact that you’ve got a boarding pass for the second sector indicates that the check has been done. Australia is one of those destinations where you can’t get a boarding pass unless it’s confirmed through system - unless they have relaxed it since I left the airline which is doubtful given how strict Australia is with this.

I suspect you are referring more to the BA document check, which if it’s all booked on the same PNR, then I believe the system will recognise the initial document check. Either way, nothing much to worry about. This would be similar to someone doing for example CDG-LHR-SYD, agent at CDG would do doc check for the entire journey.