No, this Americanism is factually wrong.
You're mixing nationality, a piece of paper, with being "native", an ethnic group. Does not work in countries across Europe, Japan, Korea, etc. The settling of the USA and its cultural dominance has broken peoples brains. Europe is inhabited by its indigenous people!
The English are an ethnic group (ask 23andMe) and they are the founding stock of Great Britain. Source - the far right extremist wikipedia: https://en.wikipedia.org/wiki/English_people
London was 97.7% white in 1960. Now it is around 50%, and falling. Again, source: https://en.wikipedia.org/wiki/Ethnic_groups_in_London
Is that good or bad? Leave that one up for debate, I mostly care about facts.
The literal definition and etymology of 'native' relates to birth, check a dictionary and the definition will read the same. The truth is that 60% of London's population is made up of native Brits and that hasn't changed at all over the last 25 years.