It is preferable to remove the ring and clean the hole before immersing in the mikveh. Therefore, it would be best to have the piercing done just AFTER using the mikveh, in order to allow about a month for it to heal before you go to mikveh again. However, if it is painful or problematic to remove the ring, and the intention is NEVER to take it out, there is room for leniency to allow immersion with this jewelry in place. The area around the ring should be gently cleaned well and, if safe, the ring should be moved around so that it is not adhering to the skin.