In chapter 2 we read: And with regard to Anne's dislike of Bath, she considered it as a prejudice and mistake, arising...secondly, from her happeneing to be not in perfectly good spirits the only winter which she had afterwards spent there with herself.

So it was perfectly acceptable for Anne to accompany Lady Russell. Perhaps Lady R. feels Bath is adequate, or at least not worth the hassle of London, since Lady R. obviously prefers Bath. Also, cost may have been an issue for Lady R; perhaps she cannot afford London (and being wiser than Sir Walter, knows enough not to spend what she can't afford!)