According to research from Imperial College London, the Met Office and the London School of Hygiene and Tropical Medicine, around 550 people died between 21 and 29 May as a result of the hot weather.
The team said a further 2,200 are thought to have died from heat-related issues between 18 and 28 June.
In the South West, the research suggested 71 people died during the May heatwave and 248 died in June.
Dr Ian McCarthy, consultant in emergency medicine at Plymouth’s Derriford Hospital, said the A&E department had been “really busy” during the heatwaves.