No, body cells die at different rates. Those with a high metabolic need for oxygen, such as brain cells, go quickly, but cells such as cartilage cells and the cells of the cornea of the eye, both areas which do not have much access to blood-borne oxygen, can live for many minutes after the heart stops beating.
In many cases organ donation is done after cardiac death, which means that the heart has stopped beating for several minutes when death is declared. As long as the body can be gotten to an operating room quickly, numerous organs can be harvested while still alive and functional. These organs are then chilled to slow the cells' metabolic rate, and can be held this way for up to several hours prior to being implanted in a recipient.
all cells do not die simultaneously, except perhaps in a nuclear explosion. Nervous cells in the brain are particulary vulnerable to oxygen deprivation and will die within 3-7 minutes after complete oxygen deprivation.