How long does the brain keep working after you've been fatally injured somewhere other than the brain?

Hard to tell: The brain heavily relies on adequate blood supply. When the blood supply is cut off, brain death will eventually ensue. It usually takes around 5 minutes for irreversible signs of brain death to develop in most cases. I am sure there might be some exceptions to this rule.

Difficult to answer: If the brain getting continuous blood flow, it keeps functioning, but it depends on status of other vital organs such as lung, heart, kidneys etc.