"How long after invading can a species still be said to NOT be native?"

That's strictly matter of opinion, and there's no way of proving who has the final opinion on it.

As regards human beings, my opinion is that you're automatically native to any area of land you happen to be born on. People are people, and they're all native to the land they are born upon.

As regards introduced animal species, imported insects, plants, etc...well, that's a bit trickier. You have to consider how they'll react with the local flora and fauna.

People did no favors to Hawaii, for example, when they brought dogs, cats, pigs, rats, mongooses, and mosquitos...all of which proved disastrous and reproduced uncontrollably. They did no favors to Australia either when they introduced rabbits.

But who ever thought ahead about it back in those days? They only found out afterward.

Raptor, you're welcome. I just expanded the borders of the topic a little, didn't I?

It's true that an adult cat does not remember having its claws removed in infancy. Neither does a human being remember its fingernails and its sexual parts being removed in infancy. Maybe we should start doing that too, and keep people indoors all the time where they won't kill things. Think how the natural environment would benefit as time went by...