I got my ink done years before I started grappling. Though I can say it took about two weeks for me to be comfortable with it being fully healed.
Think of it like any other scrape/shallow cut injury, it bleeds, and it scabs over, when that stops its pretty much good to go. Keep it clean and bandaged up like instructed and down time should be minimal.
Isshinryu Karate 2nd degree Black belt.
Neko-Ryu Goshin Jitsu 2nd degree Black belt.
Jujitsu 2nd degree Black Belt.
Isshin-Jujitsu 2nd degree Black belt.