When exactly did Voldemort start sharing his body with professor Quirrell?
Quirrell makes it pretty clear that it was after the failed break-in at Gringotts.
“He does not forgive mistakes easily. When I failed to steal the stone from Gringotts, he was most displeased. He punished me… decided he would have to keep a closer watch on me.…”
Quirrell’s voice trailed away. Harry was remembering his trip to Diagon Alley — how could he have been so stupid? He’d seen Quirrell there that very day, shaken hands with him in the Leaky Cauldron.
Harry could shake hands with him, meaning no Voldemort. Later, his scar hurts for the first time when he looks at Quirrell at the opening feast. Ergo, it was some time in the interim.
He started after Professor Quirrel failed to steal the Philosopher's Stone from the Gringotts vault. From chapter 17 of Harry Potter and the Philosopher's Stone:
‘I met him [Voldemort] when I travelled around the world. A foolish young man I was then, full of riddiculous ideas about good and evil. Lord Voldemort showed me how wrong I was. There is no good and evil, there is only power, and those too weak to seek it … Since then, I have served him faithfully, […] When I failed to steal the Stone from Gringotts, he was most displeased. He punished me … decided he would have to keep a closer watch on me …’