The father-of-three was reportedly found dead on Friday at his rented house in Salisbury, Wiltshire, by his housekeeper, the Daily Mirror reported.
Police were called to the house but say there are no suspicious circumstances and an autopsy will be carried out in due course.
Wiltshire Police said the coroner was awaiting results of toxicology tests.
Shaw was reportedly suffering from long-term health problems, including a chronic kidney disease.
He was in good spirits after returning from a recent holiday in Cuba but his friends were told that he had only 18 months to live, according to the Mirror.
Shaw and Curzon were together for six years, four of them as a married couple.
"Cressida was five when they got together and 11 when it all went wrong so Christopher would have been a strong presence in her childhood life," a friend was quoted as saying by the daily.
Shaw was married first to banker Loel Guiness' niece Liza Gard, then to a Frenchwoman.
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