Visible crew/equipment: When the maid first opens the door to Mary Poppins, you can see a shadow of something, presumably the camera, on the maid's left shoulder until she steps right.
Visible crew/equipment: As Mary walks back to sit down the shadow of the camera is on her back.
Visible crew/equipment: As all the sweeps come out of the chimneys, the wires holding them are visible.
Visible crew/equipment: As Mary Poppins gets to the top of her exit from the chimney, the wires holding her are visible. And as Bert lands, his wires are visible.
Visible crew/equipment: As Mary Poppins walks over to put the hat stand in place, the shadow of the camera plays on her dress.
Chosen answer: They recorded her singing in two separate parts and fixed them together. The editors did a good job on it! It may have been made a long time ago but even then the technology was quite advanced enough to carry out that scene.