For the record, I'm an Australian who lives in NYC. I'm very familiar with the policies of both countries.
Australia has some backwards format-shifting laws, prohibiting ripping DVDs under all circumstances for example, so it's inaccurate to pain Australia as better than the US in that regard. We can rip VHS though.
Basically, it's illegal to upload and distribute stuff, or to be making money off ripped items. If you just have stuff ripped for yourself, they are not going to care. If you're really concerned, put it all on a harddrive. If you're really, really concerned, encrypt that harddrive. If you're really, really, really, really concerned upload it and download it later. Internet speed is pretty fucking fast here.
Of course, having gone through customs numerous times with hundreds of burned DVDs, I don't think there is much cause for concern. I'd be much more worried about the UK.