There is a rabbinic decree, made at a time when mikvaot were used widely for the purpose of taharah, not to wash the entire body at once after using the mikveh. The mikvaot were not as clean as they are now 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.
Ashkenazi practice is for women immersing today to follow this decree and not shower at the mikveh after immersion. Some authorities extend this decree to not showering until after marital relations, or at least until the couple has had some physical contact, but we follow the view that the decree includes only bathing at the mikveh.
Assuming you follow Ashkenazi practice, you should not shower before reaching home. Since you may wash one body part at a time at the mikveh, you may for example just wash your hair with the shampoo that you like. However, if women may be waiting to use the preparation room when you are done, or if the mikveh attendant is waiting specifically for you to come out, it is appropriate to finish up fairly quickly after immersion.
Updated 13 September, 2023