Lawyers acting for Mohammad Asghar have called on Prime Minister David Cameron to act now to bring him home after reports of the official investigation into the shooting appeared in the Pakistani media.
Mr Asghar, from Edinburgh, was sentenced to death in January after being convicted of blasphemy.
The 70-year-old, said to suffer from paranoid schizophrenia, was shot and injured in Adiala prison in Rawalpindi last month.
Aamer Anwar, the family’s solicitor, said repeated demands by the British Government and the British High Commission for the findings of the official inquiry into the shooting have been unsuccessful.
However, media reports in Pakistan claim that the prison guard who shot Mr Asghar was incited to do so by Mumtaz Qadri, a policeman facing the death penalty for murdering Punjab governor Salmaan Taseer.
The prison guard spent more than two weeks guarding Qadri, according to reports.
Mr Anwar said: “The results of this inquiry are further proof of the Pakistani authorities’ dangerous inability to safeguard Mr Asghar’s life.
“His family are ‘heartbroken’ at what they read this morning and terrified for their father.
“They are growing increasingly frustrated at the repeated failure by the British Government to achieve a positive outcome.
“The family were promised last week that that the PM would speak to Nawaz Sharif on Friday but the call did not take place.”
He said they wrote to Mr Cameron on Friday asking him to personally intervene.
Mr Anwar said:
“Every minute that Mr Asghar spends in Pakistan jeopardises his life as well as those seeking his release. The PM must act now and the Foreign and Commonwealth Office needs to stop trying to silence the Asghar family and concentrate on bringing Mr Asghar home.”
On October 17 Mr Asghar’s daughter, Jasmine Rana, travelled from Edinburgh to present a 70,000-signature petition to Downing Street calling for Mr Cameron to intervene in her father’s case.
She and Mr Anwar had hoped to meet Mr Cameron but instead held talks with with Foreign Office Minister Tobias Ellwood.
Mr Asghar was sentenced to death in January after he was found guilty of writing letters to a number of people claiming he was a prophet.
The grandfather was arrested in 2010 following a complaint by a tenant with whom he was having a dispute.
Insulting the Koran or the Prophet Mohammed can be punished with life limprisonment or death in Pakistan, and those accused of blasphemy are at high risk of attack.
Mr Asghar has filed an appeal against his sentence but it may take several years to reach court.