A report by the Police Investigations and Review Commissioner (PIRC) said officers were sent within 35 minutes on 19 April 2016, but were then diverted.
The commissioner said the delay was caused by a lack of ownership of the initial call within Police Scotland.
The report said the man was expected to make a full recovery.
The incident happened when a City of Edinburgh Council neighbourhood support officer phoned Police Scotland with concerns about the 72-year-old who has regular welfare visits but had not been seen for a few days.
There had also been no reply at his home.
The PIRC report found while the initial call to police was correctly prioritised as a Grade 2 incident, which requires a response within 15 minutes, officers were not sent for 35 minutes.
On the way to the call, those officers were diverted to another incident and it was not until some time later that it was realised that no-one had attended and other officers were then dispatched.
When those officers arrived at the property later in the day, they found the man was seriously ill and he was taken to hospital for treatment.
The commissioner, Kate Frame, said: "Fortunately in this case, the medical opinion suggests that the delay in the police attending is unlikely to have had a significant impact on the seriousness of the man's illness.
"Police Scotland has since carried out a review of the circumstances and put new safeguards in place for dealing with this type of incident, which requires supervisors to more closely monitor the actions of staff."
Police took five hours to arrive at the Edinburgh home of a "seriously ill" 72-year-old man after a call of concern about his welfare, a review has found.