I had never thought of Jean Simmons as British either, Poppa, But I think she, like other good actors such as Meryl Streep, was good at adopting other accents called for by various roles.
I never thought of Vivien Leigh as sounding British either, but that's mainly because I remember her most as Blanche DuBois or Scarlett O'Hara.

One of the best things about the Guys And Dolls movie was the director's decision to have Brando and Simmons do their own singing, in their non-professional-singer voices. They both could carry a tune but sounded pretty much like you'd expect a gangster and a Salvation Army worker to sound.   I loved it.