Jennifer Jones, 46, was told that she had unlawfully abducted her children from their father, with whom they lived in Majorca, and that they must go back immediately.
But she evaded police by disappearing from her Welsh home in the middle of the night on Monday. It is thought that she is with the children – aged between eight and 14 – and her new boyfriend.
High Court judge launched a nationwide hunt for the children as he warned he was ‘very concerned’ about them.
Miss Jones has been in a legal battle with her estranged husband, Tomas Palacin Cambra, over the care of their children since she left him in 2008, and has abducted the children on a previous occasion, in 2009.
He was granted custody of the children, but she brought them back from Spain to Wales and made an application to the High Court to keep them. Last week a judge rejected the request and ordered Miss Jones to return the children to Spain by midnight on Friday, but she failed to do so.
She was then ordered to appear at a hearing on Monday, at which she appealed against the decision, but was told that the children had to go back immediately.
But when police officers and social workers went to her home in Llanelli in the early hours of yesterday, they found that she and the children – Jessica, 14, Tomas, 12, Eva, nine, and David, eight – had vanished.
Mr Justice Roderic Wood yesterday said at a hearing in the Family Division of the High Court in London: ‘I am very concerned about the children.
‘They must be found. I ask anyone who knows where they are or might have seen them to tell this court or the local police.’
Lawyers representing Mr Palacin Cambra, 52, said Miss Jones, who taught English as a second language, was thought to be with her boyfriend, John Williams.
She has a strong Welsh accent, is slim, about 5ft 7in tall and has strawberry blonde wavy hair and brown eyes. Mr Williams is in his late 40s or 50s and is said to be well-built.
Mr Justice Wood said the youngsters were ‘vulnerable’, adding: ‘I do not know the mother’s intentions.’
Mr Palacin Cambra, a lieutenant colonel in the Spanish army, said last night: ‘All I want is for my children to be returned safely home where they belong.
‘I am very worried for their well-being. Their lives are here in Spain, their friends are here, they are very happy here.’
Mr Palacin Cambra said he last saw Jessica, Tomas, Eva and David in June when he took them from Majorca to drop them off with their mother, under the terms of a custody arrangement.
He later travelled to the UK to pick them up but Miss Jones did not show up with the children.
All four missing children were born in Britain after the couple met in the early 1990s.
They have another daughter, Sara, 16, but she returned voluntarily to Spain in September and is not missing.
Last night Miss Jones’s family and friends said she had been driven to running away with her children as a way of protecting them.
Her mother, Brynelda Jones, said her grandchildren were ‘petrified’ of returning to their father in Spain and had begged not to be sent back.
When the children were told this weekend that they had to return to Spain two of them tried to run away, the family said. They added that they did not know where Miss Jones was with the children.