Tyson Haynes, 53. Brooklyn.Nicole Newman, 23, of south-east London, was hit when she pushed her eight-month-old baby into a pram.
Ms. Newman died at the scene on Croydon Road Peng, South East London, 13 January 2019.
Baby Luciano Bianco-Newman was seriously injured and died seven days later.
It was alleged that Haynes was traveling at a speed of 30 miles per hour at the time of the collision.
The court heard that the accident took place on a residential road near shops and houses.
On Wednesday, Haynes died of reckless driving during a brief hearing in Old Bailey.
He denied alternative charges of causing death by dangerous driving.
Prosecutor Alan Blake accepted the request after careful consideration and consultation with the victim’s family.
He said it was in the “public interest” in light of the defendant’s medical condition.
Haynes, who is out on bail, filed his petitions with the court via video link.
Judge Richard Marks said it would not be appropriate to sentence him if he was not present in person.
He warned that immediate imprisonment was a “real option”.
The victim’s family watched the action via video link.
The court heard that Ms. Newman’s partner, Luciano’s father, wanted to appear in court but was in a wheelchair due to a recent accident that resulted in a spinal cord injury.
Judge Marx adjourned the sentence without setting a date, saying it was important to deal with it as soon as possible.
Additional reporting by the Press Association