Mazal tov on your upcoming wedding!
Nails do not have to be cut extremely short; it is sufficient to cut them short enough that if you turn your palms up, you do not see them extending above the fingers. However, some kallot do wish to keep their nails a bit longer to have them manicured for the wedding. On a one time basis, you would be permitted to keep your nails slightly longer. You must take care to
clean them extremely well. You should explain to the
mikveh lady that you received a
heter to keep your nails slightly longer for the wedding.
B'Hatzlacha!