Proud Karolina Dabrowska, 36, had son Sebastian in the vehicle and even posed for pictures with the tot on the back seat minutes later.
But their joy turned to anger when dad Matthew Errington, 43, was asked by the Addison Lee driver to be paid a fee for clearing up.
Merchandiser Matthew says the driver then banged on the ambulance doors and demanded the proud father either clean up himself or hand over cash.
Matthew said: “The actions of the taxi driver at the most stressful moment of my life was appalling.”
The couple had ordered the minicab from their home in Peckham, south east London, to University College Hospital, in Fitzrovia, central London, after she went into labour.
But Karolina’s waters burst and the driver pulled over outside Aldwych One hotel, near Covent Garden.
Hotel staff gave Matthew towels and a guest, who happened to be a doctor, helped deliver Sebastian after a birthing complication.
Matthew said: “She said the baby’s coming, we need to stop. So we pulled over. I dialled 999 for an ambulance and ran into the hotel.
“There was a doctor in there and two minutes later my son was born in the back of the taxi.
“My baby came out with his cord wrapped around his neck and at that point the taxi driver starts asking me for money.
“I said my son’s not breathing leave me alone. The doctor came out and saved my son’s life.
“Everybody was so helpful, however the taxi driver saying, ‘How am I going to get my taxi clean?’
“I got my son breathing and he is still asking for money.”
Sebastian Errington was born at around 12.30am on October 25 and Matthew paid the driver around #40 for the fare, hoping that no “fouling charge”, which can be up to #100, would be ignored.
But Matthew said: “He started banging on the doors to get into the ambulance and said get out and clean the taxi.”
The pair, who live with youth worker Karolina’s eight-year-old daughter and plan to marry in Summer 2017, say they will not use the minicab firm ever again.
The huge taxi firm have apologised profusely for the horrific treatment, and given them a full refund.
Addison Lee chief executive Andy Bolan wrote to the family on Monday to apologise for his driver’s behaviour, refunded the fare and made a #100 donation to Bliss, a premature baby charity.
A spokesman from Addison Lee said: “We would like to wholeheartedly apologise again for the service that Matthew and Karolina experienced, at what was a stressful time for them.
“As soon as we investigated what had happened we refunded their fares for both journeys to and from hospital, discussed with the driver how he could have been more sensitive to Matthew and Karolina’s needs at a difficult time and are issuing guidance to all our drivers to ensure this unfortunate situation can never happen again.
“As a gesture of goodwill we have also made a donation to the premature and sick babies charity, Bliss.”