Adam Busby, 64, originally from Paisley, had been told by a judge in Dublin last July he would have to return to Scotland to face the charges.
The founder of the Scottish National Liberation Army has fought the extradition attempts since being arrested over seven offences in 2010. He has argued that, as he had lived in Ireland for 30 years, he should be tried there.
He has now appealed to Ireland's Supreme Court against the extradition order. A spokesman for the court said no hearing date had yet been set. It could take up to six months for his case to be heard.
It is claimed Mr Busby, who is in custody in Dublin, made hoax calls to media groups in Edinburgh and Glasgow and to the Samaritans. He allegedly threatened "to contaminate the drinking water supplies of major English towns and cities with a noxious substance".
It is further alleged he told a newspaper that packages containing caustic, poisonous or other noxious substances had been sent to various political leaders including Mr Brown, who was Prime Minister at the time.