The High Court was told that a notice of appeal was filed with Indonesian officials earlier this week and she was given a 14-day deadline to file grounds of appeal.
Sandiford's lawyer Aidan O'Neill QC said there was "no prospect" that competent counsel would be appointed to represent Sandiford on appeal without the Government providing some funding.
He told Mrs Justice Gloster and Mrs Justice Nicola Davies that the FCO had unlawfully fettered its own discretion by applying a blanket ban on providing legal representation to British nationals overseas.
The refusal to assist Sandiford was a breach of Government obligations to take all reasonable steps to ensure that her "inviolable human dignity" was respected and protected.
O'Neill said the Government was failing to protect her right to life - and not to face the death penalty - despite being required to do so by the European Convention on Human Rights.
FCO lawyers are arguing that it is not unlawful or irrational under UK-EU law and human rights legislation for the Foreign Secretary to refuse to fund Sandiford's appeal.
A British grandmother facing death by firing squad in Indonesia should be entitled to support from the British government because her life is at risk due to her poor finances, a court will hear.
Urgent court action is being brought against the Government today over funding for legal representation for Lindsay Sandiford, sentenced to death by a court in Bali for smuggling 4.8kg of cocaine onto the island.
The law firm involved in the case says the High Court challenge is against a decision not to arrange "an adequate lawyer" for Sandiford.
She was accused by the court of damaging the image of Bali and received the sentence despite prosecutors only asking for a 15-year jail term.
Sandiford, who is said to have no money, has notified Indonesian officials that she intends to appeal against the ruling.
Law firm Leigh Day, which is working with the charity Reprieve - which seeks to protect the interests of prisoners worldwide - said it would cost around £2,500 to pay for an adequate lawyer to take on her case.
Leigh Day is seeking a judicial review of the Government's decision not to pay.
Sandiford had not been properly represented since her arrest at Bali airport in May last year, when customs officers found the drugs sewn into the lining of her suitcase, it said.
Richard Stein, partner in the human rights team at the firm, said: "The Government has a duty to ensure that the human rights of British citizens are protected and that those sentenced to death, or suspected of or charged with a crime for which capital punishment may be imposed, have adequate legal assistance at all stages of the proceedings.
"This judicial review will challenge the Government's refusal to fund the £2,500 in expenses it would cost for a qualified Indonesian lawyer to represent Lindsay in her appeal against execution by firing squad which will take place on the beach in Bali if the Government do not act."
The Foreign and Commonwealth Office has said that the Government does not fund legal representation for British nationals abroad, but Sandiford's case was being raised through diplomatic channels.
A spokesman said: "We strongly object to the death penalty and continue to provide consular assistance to Lindsay and her family during this difficult time."
The application will be heard by Mrs Justice Gloster and Mrs Justice Nicola Davies.
On Wednesday, Sandiford's sister Hilary Parsons said she had spent the last pennies of her savings to try and find Sandiford a lawyer to challenge her death sentence for drug smuggling, but cannot afford even the expenses of an Indonesian lawyer who she says has agreed to act for free.
I can’t imagine what we will do if we miss the opportunity to save Lindsay’s life," she said.