Mr Justice Keehan said Ruby McKay-Uhd and her mother, Victoria McKay, 43, are believed to be living in a camper van, and he is “extremely concerned” about the girl.
The judge made an appeal for help at a hearing in the Family Division of the High Court in London on Tuesday.
Ruby has been at the centre of family court proceedings in Melbourne, Australia, involving her mother and father, yoga teacher Peter Uhd, 57, the judge has heard.
She was wrongfully brought to the UK by her mother in September and a family court judge in Melbourne ordered her return to Australia.
The judge heard that Mr Uhd is Australian, Ms McKay was born in Birmingham but had been living in Australia for more than 20 years, and Ruby was born in Australia.