Lindsay Sandiford, 57, from Cheltenham, was convicted last year of trafficking drugs into the resort island of Bali.
She faces death by firing squad.
The court dismissed an appeal over UK government policy not to fund the cases of Britons facing criminal charges abroad, but called on the secretary of state to look at her case "urgently".
The grandmother was found with cocaine worth an estimated Â£1.6m when she arrived in Bali on a flight from Bangkok, Thailand, in May 2012.
Sandiford claims she was forced to transport the drugs to protect her children, whose safety was at stake.
The UK Government's policy is that it does not provide funding for legal representation to any British national who faces criminal proceedings abroad - even in capital cases.
Five judges at the Supreme Court ruled the policy was not irrational and backed previous rulings by the High Court and Court of Appeal.
The Supreme Court said Sandiford "now requires a substantial sum to pay for the legal assistance" to help reopen the case and begin a "clemency petition to the President of Indonesia".
The papers need to be lodged by 29 August.
The Supreme Court's written ruling said "the local courts seem to have ignored the substantial mitigating factors in her case".
The judges cited age, mental problems, lack of previous criminal record and co-operation with police.
They said there was a "remarkable disparity of her sentence with those members of the syndicate whom she helped to bring to justice".
"On the face of it, there is substantial material to support her application to the [Indonesian] Supreme Court or the president. She needs a competent lawyer to present it," the judges said.
Announcing that the UK Supreme Court "unanimously dismisses the appeal", Lord Carnwath added: "However, in the light of new information - not available to the lower courts - as to the course of the proceedings in Indonesia and the steps now available to her there, the court calls on the secretary of state urgently to review the application of the policy to Mrs Sandiford's case in the light of that information."
The UK Supreme Court has called on the British government to review the case of a grandmother facing execution in Indonesia on drug charges.