What would be the consequences of severing the spinal cord from the brain?

The short answer is that it depends on the type of injury and where it's severed.  It could range from paralysis to death.  Please see below for further reading for understanding if you wish.

Spinal cord injury (SCI) occurs when the spinal cord is damaged in a way that results in some loss of sensation and motor control. The spinal cord is a thick length of nerve tissue that extends from the base of the brain, down the back, through the spinal column. The spinal column is made up of bones called vertebrae that protect the spinal cord. The spinal cord is made up of motor and sensory nerve cells called neurons. The motor nerves are grouped together and transmit motor commands from the brain to the muscles and initiate movement. The sensory nerves are also grouped together. They carry information about sensations, such as pain and temperature, to the brain. The spinal cord is divided into four areas: cervical (neck), thoracic (chest), lumbar (lower back), and sacral (tailbone). Each area is referred to by its first letter (C, T, L, or S) and the vertebrae within each area of the spine are numbered as follows:

C1 to C8: Cervical vertebrae
T1 to T12: Thoracic vertebrae
L1 to L5: Lumbar vertebrae
S1 to S5: Sacral vertebrae
The nerves enter and exit the spinal cord through the small openings between each vertebra. The nerves that enter through the tailbone (coccyx) are called coccygeal nerves. Spinal cord injuries occur most often in the areas of C5 to C7, and T10 to L2, where the spine is most flexible. When injured, each spinal area has distinct symptoms that help doctors pinpoint the nerves that are damaged.

Who gets it?
Approximately 10,000 people injure their spinal cord in the United States each year. Spinal cord injuries can happen to anyone, at any age. However, men between the ages of 19 and 26 are more likely to have a spinal cord injury due to an accident or some act of violence.

What causes it?
Damage to the spinal cord is usually caused by one of five different types of injuries. Concussion of the spinal cord occurs when a sudden, violent jolt injures the tissues around the cord. This injury is usually temporary and goes away with in a few hours. A spinal contusion (bruise) causes bleeding to occur in the spinal column. The pressure the bleeding puts on the spinal cord can kill neurons. Spinal compression occurs when an object, for example, a tumor, puts pressure on the spinal cord. When the spinal cord becomes torn by some type of injury, the neurons are also damaged. Or, injury occurs when the spinal cord is completely cut. Fifty percent of all spinal cord injuries occur in a motor vehicle accident. Twenty percent occur from a fall, 15% from some act of violence, and 14% from a sports-related accident. The accidents that cause many spinal cord injuries are related to drug or alcohol abuse. Other causes of spinal cord injury include infection or disease, such as cysts on the spinal cord, multiple sclerosis, or cervical spondylosis (degeneration of the disks and vertebrae in the neck).

What are the symptoms?
Damage to the spinal cord interrupts the signals from the sensory and motor nerves, resulting in varying degrees of paralysis and loss of sensation. Symptoms depend upon the severity and location of the injury. A completely severed spinal cord causes paralysis and loss of sensation below the severed section. Patients with the cord partially severed may have some function below the injury. Damage to the front portion of the cord causes paralysis and loss of the ability to feel pain and changes in temperature. Damage to the center of the cord may paralyze the arms, but not the legs. Damage to the right or left half of the spinal cord causes paralysis on the side of the injury, loss of pain and temperature sensation on the opposite side of the injury, and loss of the ability to sense position. Injuries located in vertebrae C1 to C5 may cause paralysis of the muscles used for breathing and all arm and leg muscles. Theses types of injuries usually result in death. Damage above the T1 vertebra affects the arms and the legs. Injuries below the T1 vertebra affect the legs and trunk below the injury, but usually do not affect the arms and hands. Paralysis of the legs is called paraplegia. Paralysis of the arms and legs is called quadriplegia. Complications of spinal cord injuries include deep vein thrombosis, in which lack of normal blood flow to the paralyzed area causes blood to pool in the deep veins and form clots; pressure ulcers of the skin from staying in one position for long periods of time; joint deformity caused by the constant contraction of the muscle (contracture); abnormal deposits of bone in muscles and tendons, called heterotopic ossification; and urinary infection and hardened stool in the large intestine and rectum (impaction) from the loss of bladder and bowel control. Patients with paralysis are unable to sense when they might have an over-full bladder or bladder infection, constipation or impaction, or skin irritation. When this happens, the nerves respond by releasing high amounts of norepinephrine, which is a horm