Jamie-Leigh Kelly, 31, left a Family Assessment Centre in north-west London on Tuesday September 19 with her three-year-old daughter and new-born baby boy.
The force said the children are under care protection orders and were removed from the family assessment centre against the wishes of staff.
She was seen being driven away with the children in a blue Ford Fiesta and officers have now located the car and arrested a 30-year-old woman on Friday September 22 on suspicion of child abduction. She remains in custody.
Jamie-Leigh has links to Thurrock in Essex and most recently Havering, east London, which is why the Met is leading the enquiries.
Detective Superintendent Lewis Basford who leads the Met’s Public Protection command in East Area, said: “We are making significant progress in our investigation but we remain extremely concerned about the welfare of Jamie-Leigh and her children, one of whom is less than one-month-old.
“Jamie-Leigh, I continue my direct appeal to you to contact with us so we can ensure the children are safe and well. We know you have no intention for the children to come to harm, so we are asking you to please pick up the phone and have a conversation with us so we can meet with you. The baby may need medical care as he is new-born, so please put him first and contact us on 020 7175 0793.”
Detective Superintendent Basford added: “We believe Jamie-Leigh continues to be assisted, and we ask anyone in Essex or London or further afield who may have information to call our incident room without delay, or dial 999 with any real time sightings. If you are helping Jamie-Leigh, it is not too late to do the right thing by the children and contact us.
“My thanks go to my team of detectives and our colleagues at Essex Police, as well as both Thurrock and Havering local authorities who have been extremely supportive, as we continue efforts to locate the family.”
On Wednesday 20 September, a man in his 50s was arrested in connection with the investigation on suspicion of child abduction and conspiracy to kidnap. He remains in custody in Essex after a warrant of further detention was granted by the court.
There is no indication that the children have been harmed, but detectives are clearly extremely concerned about their welfare. The baby boy was born in mid-September and so as a new-born is extremely vulnerable.
Jamie-Leigh Kelly is described as white with green eyes, of slim build and 5ft4ins tall. She was wearing a white jumper or long sleeved t-shirt, white jogging style trousers and white trainers.
Please contact the Met Police incident room on 020 7175 0793 with information or call 999 immediately with live sightings.
Alternatively, information can be reported 100 per cent anonymously to the independent charity Crimestoppers on 0800 555 111. They never ask for personal details and they do not trace your device.