Mohammed Shabir, 45, of Alice Street, Keighley, was on remand at HMP Leeds when he became unwell on 24 September.
He was one of three men due to go on trial later this year accused of murdering Bryonie Gawith, 29; Denisty Birtle, nine; Oscar Birtle, five; and 22-month-old Aubree Birtle.
The family died after a blaze at their home in Westbury Road, Bradford, in the early hours of 21 August last year.
At an inquest on Wednesday, coroner Oliver Longstaff said: “At the time of his death Mr Shabir had not been convicted of any criminal offence, he had been charged with an offence, he had indicated a plea of not guilty to that offence.
“There was a trial date set … Mr Shabir died before that trial had taken place.”
The inquest heard on the day of his death that Mr Shabir was visited by family members, who said he was his “usual self” but noted that he had become overweight during his time in prison.
CCTV footage after the visit shows Mr Shabir breathless and apparently struggling to walk up stairs, the coroner said.
He collapsed on his wing at 4.45pm and an ambulance was called three minutes later, Mr Longstaff said.
Mr Shabir was taken to Leeds General Infirmary, where his condition deteriorated and he went into cardiac arrest before going in for surgery. He was pronounced dead just before 9.30pm, the inquest heard.
The coroner said Mr Shabir had asthma and his family had concerns over whether he was receiving the correct treatment in prison.
Reading from a report by coroner’s officer Emma Walker, Mr Longstaff said: “They believe had he not been in prison he would still be alive.”
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The report said a review of Mr Shabir’s medical records in prison showed there were no noted exacerbations of his asthma in custody, he had regular asthma reviews and medication was prescribed to him regularly.
The coroner recorded a conclusion that Mr Shabir died from natural causes.
Sharaz Ali, 40, and Calum Sunderland, 26, of Calton Street, Keighley, are still due to go on trial next month at Bradford Crown Court.
They deny murdering Ms Gawith and the three children, and attempting to murder and attempting to cause grievous bodily harm to Ms Gawith’s sister Antonia Gawith.
