The answer is that there is no real time where introduced species are considered native. If they were introduced by man, they are always introduced.
We still consider species such as wild horses (wild mustangs) to be introduced (feral) animals, even though they were brought across to North America by the Spanish in the original European settlement of North America, over 400 years ago. The same is true of feral pigs.
With birds, there are lots of examples of introduced species that freely breed in this country - House Sparrows, European Starlings, Rock Pigeons (=city pigeons).
The dingos of Australia are another example. They were introduced to Australia many hundreds (thousands?) of years ago by man and now represent a "species" of feral dog found nowhere else, but they are still regarded by most as an introduced species, not a native species.
By comparison, there is another species, the Cattle Egret, that clearly introduced itself to North America after colonizing the new world without the help of man. They are not considered introduced since they got here "naturally" without the intervention of humans.
So it seems (quite rightly) that an introduced animal is always an introduced animal.
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Chris Harrison