The happy couple then posted pictures of Ms Dabrowska and her son in the back of the taxi on Facebook, the Daily Mail reported.
But their joy turned to anger when Mr Errington was asked by the driver to be paid a 100 pound (NZ$233) fee for clearing up.
Mr Errington paid the driver around £40 (NZ$93) for the fare but soon after the taxi driver banged on the ambulance window asking Mr Errington to "clean the taxi", the Daily Mail reported.
"My baby came out with his cord wrapped around his neck and at that point the taxi driver starts asking me for money."
The couple who plan to marry in 2017, say they will not use the minicab firm ever again, the Daily Mail reported.
Since the incident the the taxi firm have apologised and given the couple a full refund.