What happens when the brain is no longer capable of co-ordinating or regulating the heart and breathing rates?
Damage to the brainstem - the part of the brain responsible for controlling breathing and the heart - has different effects on the heart and the lungs.

Every heartbeat is not directed by the brain. Instead, the brain has a "throttle" to the heart, by which it can regulate the heart rate up and down, but the actual heartbeat is initiated by special cells in the heart, called pacemaker cells. In the absence of signals from the brainstem, the heart will continue pumping; this is (one of the reasons) why people who need ventilators to breath for them do not also need their heart pumped for them.

Breathing, however, is directly controlled by the brain. Without signals from the brain, breathing will stop completely (the technical term is "respiratory arrest", but that literally just means "breathing-stuff stopped-ness"). Survival time without breathing depends on a number of factors, but at room temperature, without medical intervention (ie. rescue breaths or a ventilator) and barring any special circumstances, is only a few minutes. Permanent brain damage sets in a few minutes later, even if the person can be resuscitated.

Of course, without any breathing, the heart will stop beating due to a lack of oxygen before long. The general progression will be from sinus ("normal") rhythm, to ventricular tachycardia, to ventricular fibrillation, to asystole ("flatlining").