Q. A woman that has false nails that are made to last for many months as much as half a year, does she have to remove them before she goes to the mikvah?

A. Pischey Halocho (2: p.293) writes that a woman should not immerse in a mikva if she wears artificial nails, as they are a foreign substance and should be removed prior to the tevila. Furthermore, even natural nails must be completely trimmed before immersion.
However, after the fact if she already went to the mikva, Horav Shlomo Miller’s Shlit”a opinion is that in a case of great duress where she may find it difficult to repeat the tevilah, she may rely on the first immersion since the false nails are after all decorative. (See also Pischey Halocho ibid., Chut Shani 198: 23.)

In a case where a woman is in the process of becoming a baalas teshuva and she may not go to the mikva if she is requested to remove her artificial nails a competent Rabbi should be consulted.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a