Lisa Dunn, the sister of Nick Dunn, expressed concerns that the men's mental state is beginning to deteriorate and called for action to secure their release.
"Nick's always maintained that he feels abandoned and betrayed by the government and the country that he once served," she told The Guardian.
"Now we have a new government in place I would like to personally reach out to Theresa May, Boris Johnson (foreign secretary) and Sir Alan Duncan (a Foreign Office minister) and respectfully request that they continue to keep this case at the top of their agendas as we have been assured many times previously," she said.
They were held while working for an anti-piracy security company protecting commercial ships off the coast of Africa in the Indian Ocean.
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The men, who have been backed by more than 20 British MPs, including former British PM David Cameron, have consistently maintained their innocence and launched an appeal to overturn their sentences.
A petition calling for their release has garnered 375,000 signatures and was delivered by the families of the six former soldiers to Downing Street last week.
"I appreciate and understand that the government have spoken to various Indian counterparts over the last nearly three years, but for the evidence that's there it's beyond belief that our government haven't pushed harder," Lisa said.
"They keep saying we've talked with this Indian counterpart, but it was apparent a long, long time ago that talking makes no difference to the Indian authorities. We need more robust action," she added.
She has also claimed the men were suffering in the Indian jail and have to sleep on concrete in cells infested with snakes and rats, using a hole in the ground for a toilet.
"We recognise what a difficult time this is for those involved. We cannot interfere with India's independent legal system, just as other countries cannot interfere with ours, but we will continue efforts to make sure this case is resolved swiftly. Ministers will continue to raise this case at the highest levels.