They are the first cases reported there since the virus spread over the border from Democratic Republic of Congo, where nearly 1,400 people have died.
Three suspected Ebola patients have also run away from an isolation centre in south-western Uganda.
The head of a major medical charity has called the latest outbreak of Ebola in central Africa “truly frightening”.
Dr Jeremy Farrar, the director of the Wellcome Trust, said the epidemic was the worst since that of 2013-16 and has shown “no sign of stopping”.
He said the spread was “tragic but unfortunately not surprising”. He warned that more cases were expected, and a “full” national and international response would be needed to protect lives.
On Friday the World Health Organization (WHO) will decide whether the outbreak should now be deemed a global health emergency.
The grandmother had lived in DR Congo, where her husband recently died of Ebola. Her daughter had travelled from Uganda, where she lives with her Ugandan husband, to the Congolese town of Beni with her children to help care for him.
After the grandfather died they all travelled on Sunday to Uganda’s Kasese district, where the five-year-old son and his grandmother then subsequently became sick and later died.
The three-year-old son is now confirmed to have Ebola. He, his parents and two of his siblings have been repatriated at their request to DR Congo, Uganda’s health ministry says.
Twenty-seven people are said to have been in contact with the three confirmed cases in Uganda. They have been restricted to their homes and will be vaccinated against Ebola.
The people who fled from a hospital isolation unit had been found to have high temperatures when they crossed the border from DR Congo to the Ugandan district of Kanungu, which is about 150km (93 miles) south of Kasese. Medical workers did not get a chance to take samples of their blood to send for testing before their escape.
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