In Mary's personal letters....

[Anita]

Jean Baker states on page 102 of "Mary Todd Lincoln- A Biography", that it could have been one of the Springfield doctors that attended Mary during Robert's birth since several advertised as specialists in obstetrics and women's diseases. She states that Mary's friend Dr. Anson Henry charged five dollars for an "accouchment" so he could have been present, or William Wallace but as her was her brother-in-law it would have been an embarrassment.

With both Mary's mother and Lincoln's sister dying from complications of childbirth I would think Mary and Lincoln would have wanted a doctor present. Lincoln blamed his sister's death on his brother-in-law for not sending for a doctor.

I agree with Roger that "Possibly the answer is lost to history."