A kallah may certainly shower after her immersion in the mikveh, though not at the mikveh site itself.
There is a rabbinic decree not to wash the entire body at once after using the mikveh, a decree that was made at a time when everyone regularly used mikvaot for the purpose of taharah. The mikvaot then were not very clean and the decree was intended to prevent the erroneous assumption that the cleansing after immersion is actually what makes one tehorah – an error that could lead people to skip mikveh altogether.
Even today, Ashkenazi custom is to follow this decree and women do not shower at the mikveh after immersion. Some women wait to shower until the couple has had some physical contact, and others extend this to not showering until after marital relations. We follow the view that the decree only applies to bathing at the mikveh, but that one may bathe or shower after returning home. Furthermore, it is permissible to wash one part of the body at a time, even at the mikveh.
Note that the decree applies only to mikveh night. If relations are delayed for any reason, the woman may shower when she returns home.
Updated 13 September, 2023