The Dublin-born Care International worker was taken hostage in October 2004 and killed just under a month later. Her body has never been found.
Yesterday, her family said that during her captivity, four calls were made to her Iraqi husband Tahseen in Baghdad from the kidnappers, demanding to speak to a member of the British Embassy.
But Mr Hassan had been told by the British that they would not speak to the hostage-takers.
“We believe that the refusal by the British government to open a dialogue with the kidnappers cost our sister her life,” Deirdre, Geraldine, Kathryn and Michael Fitzsimons said in a statement.
The charity worker had Irish, British and Iraqi nationality and lived in Iraq for 30 years.
Three men arrested by US troops in May last year and held in connection with her kidnap and murder go on trial in Baghdad today.
The family said they had begged then British Foreign Secretary Jack Straw and his successor, Margaret Beckett, as well as the British Foreign Office, to arrange for the men to be interviewed by British military police.
“They have refused this request even though this is the only way that Margaret’s remains will be found, and we can bring her home to be buried with the dignity she deserves,” they said.
In the statement, Ms Hassan’s brother and sisters said: “We believe the time has now come for the British and Irish people to know the truth of what happened to our sister Margaret, a British subject.
“During the period of her captivity, four calls were made from the kidnappers to her husband Tahseen in Baghdad. These calls were made from Margaret’s mobile phone.
“The hostage-takers demanded to speak to a member of the British Embassy, but Tahseen had been told by the British that they would not speak to the kidnappers.
“We believe the refusal by the British Government to open a dialogue with the kidnappers cost our sister her life. Margaret, who was vocally opposed to the war in Iraq, was sacrificed for the political ends of Tony Blair and George Bush.”
Britain’s Foreign Office confirmed Ms Hassan’s husband was called from her phone by someone purporting to be holding her, but said they had been unable to confirm the claims.
A British Foreign Office spokesman said yesterday they “understand her family having criticisms”. He added that officials would be closely following the trial of the three men, which begins in Baghdad tomorrow.
Foreign Affairs Minister Dermot Ahern has appointed a lawyer in Baghdad to represent the Government at the trial, his spokesman said yesterday.
During Ms Hassan’s kidnap, in which video recordings of her pleading for her life were released, officials were keen to distance her from the British government and emphasise her charity work in Iraq, as well as her Irish citizenship.