With brain injuries, it is not recommended anymore to keep patients awake because
people with a concussion need to sleep to recover. In the days before head CT scanning was widely available, the only way to know if someone had life-threatening brain bleeding (which occurs in less than 0.1 percent of those with concussion) was to observe him for a decrease in his level of alertness that resulted from the blood pressing on vital brain structures. This usually happened within six hours of injury. It was thought that if you could keep someone awake you could prevent him from lapsing into coma, which of course did not work.
While some people on the internet claim that someone awake with a higher blood pressure and a faster pulse will have better survival chances, (actually, this would mean that they bleed more, not less), there is no scientific evidence that this will indeed improve survival chances.
Emergency services usually check whether the patient is still able to stay awake to assess his situation based on the Glasgow Coma Scale (GCS). Saying "Stay with me, buddy" will then be used to asses the verbal response.
However, this is only used to assess the current life-threatening dangers (for example in the various Triage systems). Keeping someone awake will not benefit their survival chances.
Mostly, trying to keep someone awake only creates very dramatic scenes for movies.