A firearm was found near to her at the property in Norton, Wiltshire, The Sun reports.
An inquest at Wiltshire and Swindon coroner’s court was opened and adjourned until October 25.
Coroner Grant Davies said police ‘have deemed the death as non-suspicious and there was no third-party involvement’.
Ms Roche had been studying for a degree in English Literature at Durham University.
A spokeswoman added that William and Harry’s cousin ‘will be sorely missed’.
An obituary published by the Yorkshire Post said: ‘Roche, Rosie Jeanne Burke. Died on Monday 14th July 2025.
‘Darling daughter of Hugh and Pippa, incredible sister to Archie and Agatha, Granddaughter to Derek and Rae Long.
‘Private family funeral. A memorial service will be held at a later date.’
The death notice of Ms Roche appeared in The Times newspaper.
The student was a scion of the Barons Fermoy and the eldest child of (Edmund) Hugh Burke Roche, 53, and his wife the former Phillipa Kate Victoria Long.
Her grandfather was the 5th Baron Fermoy, Princess Diana’s uncle, who shot himself dead in 1984.
Last February, Thomas Kingston, Lady Gabriella Windsor’s husband, died from a head injury, with a gun found near his body at his parents’ home in the Cotswolds.
An inquest in January heard that he had been given several drugs to treat depression by a GP at the Royal Mews Surgery, a practice at Buckingham Palace, after having trouble sleeping following stress at work.
In the days leading up to his death, the former hostage negotiator turned financier had stopped taking any medication and toxicology tests showed caffeine and small amounts of sleeping tablet zopiclone in his system.
The coroner found that he took his own life while ‘suffering adverse effects of medication he had recently been prescribed’.