When it used to say UK passports only for the Immigration I used to show the British one and also state I had an Australia one also , this never perturbed any immigration officer as long as it was declared, think it is also on your landing form to declare if you wish, however some of the immigration paperwork is now done on a machine supposedly to make it quicker. Don’t think my passports were every geared up for that, or if they were never bothered , the same as I don’t bother in the super market , would rather deal with a cashier , especially if she’s a looker , who wants to talk to a machine. Cheers JS