There is no way to know for sure what nationality Miss Brill is. Katherine Mansfield does not include that detail in the text. What the reader does know is that the story is taking place in France and Miss Brill is there to teach "English pupils":
. . . and it also explained why she had quite a queer, shy feeling at telling her English pupils how she spent her Sunday afternoons.
Of course, the above quote doesn't necessarily mean that Miss Brill is teaching English. The line could be interpreted that she is teaching students who are English and living in France. She might be teaching history to English students. Or she might be teaching English (language) to French students.
We could assume that she is teaching the English language to French students, but it would then be possible for her to be English, American, Australian, or any other nationality that primarily speaks English.
I do tend to think that she is English, though. The usage of the word "queer" in the above quote is distinctly British in nature. My other reason is that the story singles out an "Englishman and his wife" at one point. Miss Brill likely has some experience and knowledge of what that nationality looks like in order to know, based solely on appearance, that the couple is English.