Eight-month-old Rose Paduraru is thought to be with her seven-year-old sister Oliwia Ilksa and 12-year-old brother Krzysztof Ilksa.
All three youngsters were living with their mother in Hillingdon, and social services staff at Hillingdon Council had responsibility for their welfare, the judge said.
Mr Justice Keehan said he was "very seriously concerned" for the children's welfare on Tuesday (November 14).
He said he was particularly worried about Rose, and urged anyone who had information about the youngsters' whereabouts to call police immediately.
All three are at the centre of family court proceedings.
Mr Justice Keehan said they had vanished a few days ago after another judge decided that they should go into council care.
He appealed for public help after analysing the case at a hearing in the family division of the High Court in London.
'Vitally important they are found as soon as possible'
The judge said Rose's mother Joanna Ilksa and father Nelu Paduraru, who are both in their 20s, had been arrested following the children's disappearance.
"I am very concerned for the welfare of these children - particularly the youngest, Rose," said Mr Justice Keehan.
"It is, in my judgment, vitally important ... that they are found as soon as possible."
He added: "I would urge anybody who has any knowledge about the whereabouts of these children to contact the police or the local authority immediately and without any delay."
The judge is due to consider the case later in the week.
Mr Justice Keehan issued an appeal after hearing from lawyers representing Hillingdon Council and Rose's parents.
Lawyers said Rose's mother was Polish and her father Romanian.
They said the couple were separated and Mr Paduraru lived in Greenford , Ealing.
Keep up to date with the latest news in west London via the free getwestlondon app.
You can even set it to receive push notifications for all the breaking news in your area