As far as I can tell, you don't need to have a fixed address to get a National Police Check. You do need to have various documents, some of them, such as a drivers licence, will have an address on them, but you can have for example a password and a birth certificate, and those have no attested home address on them.
Has anyone been able to get a National Police Check completed, or know someone who has, without a fixed address? DM is fine if you prefer.
