Minerva McGonangall encounters Hagrid early in the morning. He does not have Harry with him any more, apparently. She is not even informed at this point what the signifigance of that address is, since she is shocked later to discover that Dumbledore intends to leave Harry there and is not aware that those horrible Muggles are Harry's relatives. Hagrid also does not tell her that James and Lily were killed. She goes to Privet Drive quite early in the day in the form of a tabby cat and waits on a stone garden wall.
Celebrations continue all day.
She waits there until almost midnight Peter Pettigrew flees, pursued by Sirius Black. He is cornered in a Muggle street. Pettigrew turns into a rat and escapes into the sewer, leaving his finger behind. Sirius is blamed for betraying the Potters and for murdering Pettigrew and a bunch of Muggles. A Junior Minister in the Department of Magical Catastrophies is first on the scene.