Thank you for reaching out.
Preparation for the mikveh is not viewed as bathing for pleasure and therefore the usual preparation routine is permitted during the
Nine Days (although not on Tisha B'Av itself).
Given that the restrictions of the Nine Days only begin for Sephardim on Motzaei Shabbat, it may be preferable for you to do the bulk of your preparations on Friday and then just a shorter preparation after Shabbat before immersing.
There is debate among halachic authorities about married women shaving body hair during the three weeks, so it is best to check with your rabbi about this. Cutting nails as part of your mikvah preparations is permissible.
Please feel free to get back to us with any further questions.
B'Hatzlacha!