Mr Al Thani and his co-convicted wife Iman Al Kuwari had not yet served any of their six year prison sentences. Mr Al Thani had only attended a single hearing since the triplets' death.
In April, Prime Minister John Key met with the Emir of Qatar, Tamim bin Hamid Al Thani, to discuss the death of the Weekes triplets.
At the time, the Emir responded that he was "utterly committed to making sure those who are responsible will be held to account."
"We can only hope that the Qataris take note of this report, because it is this very system that has meant we cannot escape the horrors of 28 May 2012," Mrs Weekes said.
"We know, more than anyone, the inadequacies of this legal system, and the immediate need to change it.
"We remain very distressed that this system has failed us so far, and that our children's killers use that system to their own benefit including not even attending court."
The UN Report recommended that Qatar improve its legal systems - noting its "considerable financial capabilities" to do so. It also recommended that human rights mechanism like the International Covenant on Civil and Political Rights be ratified.
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