Addressing the situation in an interview with The Telegraph, she said: “Part of me wishes I could control it, but I can’t. I just slip into one or the other.
“When I moved to the States I tried hard to cling on to my British accent because it made me different.”
Gillian previously won a Golden Globe for best TV drama actress on The X-Files in 1997.
Her The Crown co-stars also collected awards at this year’s ceremony, with Emma Corrin, who played the role of Diana, Princess of Wales, taking home the Best Actress in a TV Series gong.
Josh O’Connor, who portrayed a younger Prince Charles, also accepted a Best Actor in a TV series award for his performance, while The Crown won best TV series.