The Officials involved have believed Quirrel to be David Monroe for a long time (chapter 84), that seems to have become the Official Story over the last couple days, and Mad-Eye Moody knew that Quirrelmort should have been fired out of a cannon into a sun as soon as he took the Defense position, so they've been asking questions and taking the first reasonable answer for a while.

I'm fully expected Moody to dive through a door with stunners flying, yelling "Not paranoid enough!" because /someone/ didn't expect door transfiguration, but most of the other people involved have already gotten suspicious and had their suspicions allayed.