I see what you did there ...
That doesnt seem to hold true for the ESL accents almost anywhere in the world, though. Most people's accents sound like their native tongue.
I really like running across a person developing an American Southern accent with their English learning.
Depends what you learn from. If you learn English from the British then sure. As you might in Europe.
But if you learn from Americans, or even from American TV, well, that's what you'll learn.
There's no one "British" accent though, people learning English in a region of England or Scotland do and should pick up the local accent. What works is correct, speaking like a local is what works.