A lot of time passed between when Kirk and Carol met and when we saw her in TWOK. It's entirely possible she did have an English accent when they met and that it changed over the subsequent 20 years to a more American-sounding accent. People's accents change all the time.
Bibi Besch, the original Carol was born in Austria and spent the first six years of her life there. No doubt she had a Austrian accent at one time.