I have an Australian acquaintance who got into trouble recently since he was born in Scotland but moved to Australia as a child. He is about as Aussie sounding as they come and went to school there, but after working in the UK for a year or two, Australian immigration gave him grief when he decided to return.
Australians actually comprise one of the biggest groups of illegal migrants in the UK, but they are more rarely commented on as they are much less visible than certain other groups. Australians with a parent or grandparent from the UK qualify, whereas those who don't, don't. But the ones who don't are rarely deported, and can't be easily distinguished in many cases.
He's not a member of ACDC is he? ;)
I think many in the UK don't really regard Aussies or Kiwis as "immigrants" at all. I have various red faced in-laws with political views that are, shall we say, to the right of centre and they see Aussies as just Brits with funny accents.
But yes, if you had someone whose UK ancestor back in the 19th century, or descended from Italians, Greeks or Lebanese, with an Aussie accent, then you're unlikely to be a suspect.
don't bring anything organic to Australia unless it's been commercially processed, and even then show it to the border officers so they can make the call. You don't get penalized for showing them something not allowed, you just can't bring it into the country, and they throw it in the bin for incineration. Penalties start when you try to sneak something in anyway.