Hello,
It can be due to a delayed allergic reaction to any of the ingredients of the tattoo ink and can be triggered by hormonal changes in body. Take some antihistaminic like Benadryl or Claritin after consulting your doctor( since you are pregnant) and apply some calamine lotion on the rash. If the symptoms persist then please get it examined from a dermatologist.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.
Could it be the body's reaction to the inks used?
Chances are it is the hormonal changes you're going through right now with the pregnancy. Just because it doesn't itch does not mean it is not an allergic reaction.
You can try an aloe vera gel to calm any inflammation and redness and speed the healing. An antihistamine can also be helpful, but make sure it is okay to take during pregnancy. Your doctor might even have a safe brand to give you on your next visit-- just don't forget to ask him this time! ;-)
Congrats on the baby!