Sorry to join late - BA staff have always been very helpful.
Sounds like you already bought a ticket but she could have done it with two CX sectors if the ground staff in SYD would have issued the 2nd boarding pass and tagged the bags all the way to LHR. Big gamble but it might have worked out.
Exit permit as a British citizen is not required.
Actually the Aussie government told every foreigner to go home months ago.
Good luck