Factual error: When the train is travelling away from Paddington Station, the locomotive does not have a number on the buffer beam, but has a number plate on the smokebox door. It also has a large 'British Railways' symbol on the tender. In 1940, the Great Western Railway would have had the number on the buffer beam and a either a small GWR logo on the tender or Great Western in full. British Railways wasn't created until 1948.
Continuity mistake: When the Pevensies are playing hide-and-seek, there's a shot of Lucy running up the stairs. If you look closely, she is wearing the dress that she wears in her third time to Narnia, not the dress that she was wearing when she goes through the wardrobe into Narnia for the first time. (00:10:55)
Continuity mistake: After Lucy pulls the cloth down, off from the wardrobe, its position on the floor drastically changes in following shots. (00:11:35)
Continuity mistake: In the White Witch's castle, when Edmund speaks with her about Aslan, his shirt collar goes from being over his sweater, to under, then over again.
Continuity mistake: After Lucy and Mr. Tumnus meet, when they pick up the scattered parcels their positions in his arms differ, and the string/bows tied round them differ repeatedly between shots.
Continuity mistake: As Jadis, the White Witch, and her army approach, and subsequently battle the army of Aslan, both army's collective shadows differ throughout this scene.
Continuity mistake: As Lucy watches the dazzling display in the fireplace, she alternately holds her cup down on the saucer/up by her face. (00:20:10)
Continuity mistake: When Peter, Susan, Lucy and the Beavers arrive to Aslan's camp and see Oreius, his body is all black. In the battle you can see that Oreius has black legs up to his knees, but then his body is brown. It's easy to see just when the Minotaur drops from Oreius's back: the stomach is brown.
Continuity mistake: When the Pevensie children leave Mr. Tumnus' home they hear noises outside, so Susan places her hand on Peter's arm and Peter holds her hand. Just before Beaver appears, in the front shot, Susan and Peter remove their hands, but in the next shot they're back on Peter's arm. (00:44:50)
Other mistake: When Aslan is resurrected, and the Stone Table cracks, you see Susan and Lucy stumble from the jolt and fall. If you watch closely, however, Susan prepares herself for the jolt before it ever happens. She throws her arms out and bends slightly at the waist, a few seconds before the jolt.
Continuity mistake: After Edmund has been rescued and is talking to Aslan on the cliff, as he starts walking back to the tents, Aslan in the background is suddenly taller than him, even though he is further from the camera. In the next shot Aslan's height is normal again.
Continuity mistake: When Edmund first enters Narnia, he is attacked by the White Witch's henchman, who pushes him into the snow. In some shots, there is a considerable amount of snow in his ear, and in others, its completely gone. This switches several times. (00:29:15)
Continuity mistake: When Lucy, Edmund, Peter and Susan fall through the wardrobe together, as they stand there the amount and pattern of snow on Edmund disappears, then reappears, and changes drastically between shots, while they're arguing before they head off. (00:40:40)
Visible crew/equipment: When Maugrim says "I won't wait forever", crew lights are reflected in Peter's sword. (01:14:50)
Continuity mistake: When Lucy, Peter, Susan and the Beavers speak with Father Christmas, the position of his large bag when it's on the ground, and then in the sleigh, distinctly differs between shots.
Continuity mistake: As Edmund lays prisoner in the White Witch's cell, the long chain dragged along the floor, binding his feet, changes position during shots. (01:03:35)
Continuity mistake: When Edmund falls backward into the snow, after his near hit by the White Witch's sleigh, his dressing gown changes from hanging down over his knees, to folded back, over his chest. (00:28:45)
Factual error: During the opening WW2 battle scene, the German pilot's boots have modern Vibram soles. (00:01:00)
Continuity mistake: Peter and Aslan are standing, looking out over Narnia and in frontal views Aslan is only as tall as Peter's shoulders. But from reverse shots Aslan is taller than Peter.
Continuity mistake: When Lucy first enters Narnia, her hair changes between shots from being straight and neat to being messy and slightly curly.
Answer: Spoiler alert: this gives some important plot twists away. Sometimes a bit of unresolved mystery improves a story, and I think this is the case here. But the book partly answers your questions. At the end of the last chapter it is shown that Mrs MacReady thinks the wardrobe is just a piece of furniture. She knows nothing about Narnia. But Professor Kirke amazes Peter, Edmund, Susan and Lucy by expressing familiarity with Narnia and explaining that a wardrobe might well be a portal into Narnia. If C S Lewis had not written any more books after completing "The Lion The Witch And The Wardrobe" Professor Kirke's knowledge of Narnia would probably have been an unresolved mystery. But C S Lewis later wrote "The Magician's Nephew" which tells how Professor Kirke visited Narnia as a boy. The final chapter of this book says he took an apple back with him, which he planted in his garden. It grew into a tree, was cut down and made into the wardrobe. So Professor Kirke was not consciously aware of what the wardrobe could do, but with hindsight, he realised that he had set up a chain of events that caused the children to discover Narnia.