Why doesn't the Doctor regenerate immediately in “The Night of the Doctor”?

I think that we can only offer speculation until the Anniversary special is broadcast (and I certainly don't know of any precedent in canon) but I note that the Sisterhood of Karn indicate that they can control the form of The Doctor's next regeneration. This suggests that his temporary restoration to life (presumably using a draft of elixir) may have actually arrested his regeneration. The second draft then reverses this, dictating his form.

The other possible explanation is that regeneration can only repair a certain amount of trauma, and there is a point beyond which a Timelord cannot regenerate unassisted (so a Timelord at ground zero in a nuclear explosion could not reassemble, for example). The death of the Doctor in "Vire a esquerda" would appear to support this possibility.

(Interesting side question, does this mean that The War Doctor is now the Ninth Doctor, and we have to renumber from then on?)

We've seen a few times the Doctor in a situation where he's close to death and can't regenerate - in Let's Kill Hitler, for example. It's not too much of a stretch to reason that plunging into the ground at high speed in a crashing spaceship rendered him much more seriously injured than, say, falling off a radio telescope, and therefore killed him outright until the Sisterhood temporarily reanimated him - but that status left him unable to regenerate until they helped him do so.

He died too quickly for his body to regenerate - ergo, the Sisterhood of Karn had to first bring him back to life before giving him the elixir for regeneration.

There are a few precedents for this in the show:

In "Turn Left", a parallel world is created around the Doctor's companion Donna, and in it, the Doctor dies as a result of his reckless genocide of the Racnoss race. When his body is found, a soldier states, "I think so. He just didn't make it out in time... the Doctor is dead. Must have happened too fast for him to regenerate." The Doctor killed the Racnoss by flooding its base, so presumably, drowning can kill a Time Lord before they can regenerate.

This is reaffirmed in "The End of Time" when the Doctor tells his friend Wilfred Mott, "I can still die. If I'm killed before regeneration, then I'm dead."

The crash he was in killed in instantly, so the Sisterhood had to revive him, so he could regenerate. Think what the Doctor said in The End of Time Part 1: "I can still die. If I'm killed before regeneration, then I'm dead" So there you go.