My mother lost her English accent the moment she stepped off the plane in New York, she said. But that's more than 50 years ago...I suspect that a linguist might be able to spot in which country she was born and raised, but the rest of us can't! She's been in the US since she was 19, so that may explain why she lost the accent, although she always said she wanted to be American and fit in, so she's done so.
Another lady in our town never lost her accent that I could tell. I always found it fascinating to listen to her when I was a child, partially because I couldn't believe she and my mother could possibly be from the same country!
Then there was the Scottish lady whose kids I used to babysit for - in those days, she'd only been in the US for a few years and I was a teenager, so it was always best for her to write things down for me, I had a difficult time understanding her. My mother tells me she died recently, but she never lost her Scottish accent...