Rebecca Richardson, 54, from Cradley in Herefordshire, was discharged from the extradition request at Westminster Magistrates’ Court last month.
Turkish authorities had sought her return over allegations she assaulted a young child while teaching at an international school in Istanbul in early 2019.
Last month Judge Paul Goldspring gave the Crown Prosecution Service (CPS) two weeks to consider whether to challenge the decision in the High Court.
But at a brief hearing on Tuesday, the CPS confirmed it would not pursue an appeal.
The court previously heard that Richardson left the UK in 2000 and spent nearly 20 years living abroad.
She lived in Turkey between 2013 and 2019, but before that had lived in Mexico and Hong Kong, the extradition hearing was told.
