In 1944, many children are evacuated from London to the countryside. The Rawlins siblings - Charlie, Carrie, and Paul - are sent to Pepperinge Eye, Dorset. They are placed in the reluctant care of Miss Eglantine Price, a reclusive, eccentric woman, who at first refuses the request til learning the foreign office's laws. Miss Price is strict yet kind, as well as secretive, living alone in her late father's house aside from her black cat Cosmic Creepers. When the children try sneaking out the house, they witness Miss Price flying on a broomstick, only to crashland.
Her secret ousted, Miss Price confesses she is an "apprentice witch", and a student of the Correspondence College of Witchcraft, headed by Professor Emilius Browne. Charlie attempts to blackmail her, so Miss Price temporarily transforms him into a white rabbit. Deciding to trust the children, Miss Price agrees to give them a travelling spell, a type of Utility Magic that lets them travel to where they want to go. She enchants a bedknob from her father's bed to perform the spell, which only Paul can use, by telling it where he wants to go and tapping it three times.
Miss Price receives bad news upon learning the college has abruptly closed down. Hoping to acquire the final spell from the course to help with the war effort, Miss Price convinces Paul to help her travel to London to find Browne. Using the bed, they fly through a Disney Acid Sequence to London. They meet Browne, who turns out to be a charismatic Con Man and Stage Magician, who disbelieved in actual magic until Miss Price turns him into a rabbit. Mr. Browne explains he got the spells from an old Spell Book in his possession. The group travel to an abandoned manorhouse currently occupied by Mr. Browne (thanks to a dud German bomb). The children explore, Paul finding a storybook called "The Island Naboombu". Mr. Browne proposes that he and Miss Price go into showbusiness as magicians, but she is more interested in finding the spellbook. After being turned into a rabbit again, Mr. Browne yields the book, The Spells of Astaroth. However, Miss Price discovers the other half, featuring the spell "Substitutiary Locomotion", is missing.
The travellers head to Portobello Road in London to find the rest of the book, but are unsuccessful. They are approached by Swinburne, a spiv, who works for a collector named the Bookman, who owns the other half of the manuscript. Miss Price and the Bookman trade their halves, but find the incantation for Substitutiary Locomotion to be missing, described as being on a medallion called the Star of Astaroth. The Bookman explains that Astaroth was a wizard, who used his magic to experiment on animals, giving them human characteristics. The animals rebelled, killed Astaroth, stole the Star, and escaped on a ship. A lascar claimed he saw an island ruled by animals — the Island of Naboombu. When Paul reveals its exsitence but refuses to share his storybook, the Bookman threatens him with a knife. The group make a getaway on the bed.
The bed materialises in the animated world of the island, landing in the lagoon and meet the local fish. Miss Price and Mr. Browne enter a dancing contest and win, only for the bed to be fished out of the sea by a dim-witted bear. He takes them to meet King Leonidas, who wears the Star of Astaroth. Mr. Browne offers to be the referee for a soccer match, much to the King's delight. The soccer match turns out a match of Calvinball, the animals cheat and make up the rules as they go along, and Mr. Browne takes a beating as he tries to obtain the Star. Leonidas declares himself the winner by blowing the deflated ball into the goal. Mr. Browne swaps the medallion for the soccer whistle. Leonidas realises this and chases the group. Miss Price turns him into a rabbit and they escape home.
However, when Miss Price prepares to use the spell, Mr. Browne discovers the star has disappeared because the world-jumping variety of the trope Can't Take Anything with You applies. Paul reveals the words have been in his storybook the whole time (since no one bothered to listen to him). Miss Price uses the spell, bringing Mr. Browne's shoes to life, but a horde of others clothes spring to life, causing mayhem. Mrs. Hobday, the village's postmistress, drops by to inform Miss Price that the children can be relocated, but she declines the offer. Mr. Browne scarpers when it comes to family commitment.
A platoon of Those Wacky Nazis sneak onto shore and invade Miss Price's house, led by Colonel Heller, to cause mischief. Miss Price and the children are locked in a nearby museum. Mr. Browne, spotting Nazis cutting phone lines, flees back to the house. He transforms himself into a rabbit, bounding to the museum to help free Miss Price. Using Substitutiary Locomotion, Miss Price animates the museum's collection of medieval suits of armour and military uniforms. The army and a flying Miss Price chase off the Nazis, but Heller bombs Miss Price's workshop, destroying her source of magic and breaking the spell. Miss Price decides to retire from witchcraft, but is proud to have contributed to the war effort a little bit.
Mr. Browne signs up to join the army and departs with the local Home Guard as an escort. While the children worry their fun is over, Paul holds aloft the bedknob, reminding them it is still enchanted.