Sections 97(2) and (6) of the Children Act 1989 make it a criminal offence for anyone to publish material likely to identify a child involved in proceedings under the Act. The prohibition only lasts until the end of the proceedings - Clayton v Clayton  1 FLR 11.
More significant is section 12(1) of the Administration of Justice Act 1960 which makes it a contempt of court to publish information relating to proceedings. In Re B (A Child) (Disclosure)  2 FLR 142, Munby J held that section 12 stops dissemination of what happened before the judge and the documents filed in the proceedings. The prohibition operates even where documents are anonymised. Such an injunction may be permanent.
Re J (Reporting Restriction: Internet: Video)  EWHC 2694 (Fam) explains that the Family Division has a broad discretion to extend and relax the “automatic restraints” imposed by these two pieces of legislation under the Inherent Jurisdiction and under section 6 of Human Rights Act 1998. Courts must follow the approach set out in Re S (Idenfication: Restrictions on Publication)  UKHL 47  1 FLR 591.
(a) Neither Article 8 nor Article 10 have precedence over each other.
(b) Where the two articles are in conflict an intense focus on the comparative importance of the specific rights being claimed is necessary.
(c) The justification for restricting each right must be considered.
In short, the Family Division can make whatever order it deems appropriate potentially against the world forever if it is proportionate.
To some such powers represent an abomination allowing secret courts to make unacceptable decisions that sometimes permanently separate children from their parents and afford those who do it anonymity.
To others, family proceedings are private NOT secret involving the discussion of sensitive information and represent the most significant intrusion into the lives of private citizens that any court can make. The court should be slow to allow publication unless the parties agree.
Whilst local authorities can obtain injunctions and bring committal proceedings in the traditional way one wonders whether the law can effectively be enforced. Those who will not voluntarily purge their contempt cannot be made to take down material online having already been punished. The sites that host them are outside of the territorial reach of the Family Division.
The High Court may have the power to make orders against the world but is anyone listening and does anyone care?