Over 4,000 people have signed a petition backing her.
She said this week that the dispute had left her sons traumatised.
“They’re still not talking and they’re still in nappies. It’s very humiliating for them. When they were taken away it was because they didn’t speak 200 words, now they can’t speak 50 words — they can’t use sentences,” she said.
Ms Alexander has hired a new lawyer, Dr Alfred Kriegler, to conduct her appeal.
She said: “My last lawyer gave up the case. I just hope now there will be an independent assessment of the boys, That’s all I want.”
Dr Kriegler said: “There are some points in the case that seem bizarre. The grounds of the appeal will be lack of impartiality of the judge and procedural mistakes.”
He added: “My client is really suffering. We will try our hardest to bring the twins back to her.”