Bert, a man of many professions, introduces the audience to Cherry Tree Lane ("Chim Chim Cher-ee/Cherry Tree Lane- Part 1") in Edwardian London. Number 17 is where the Banks family lives: George and Winifred Banks, their two naughty children Jane and Michael, their cook, Mrs. Brill and their odd-job man, Robertson Ay. Things are not going well for Jane and Michael. They are out of control and, as the show starts, their latest nanny, Katie Nanna, storms out. After that, Mrs. Brill and Robertson Ay complain about living in a "madhouse." The children decide to write the advertisement for a new nanny ("The Perfect Nanny"), but George, having a very different idea of what constitutes the perfect nanny, tears up the piece of paper and throws it in the fireplace. Within moments, Mary Poppins arrives, restored advertisement in hand, and takes charge of the Banks children, having every confidence in her own qualifications and merits ("Practically Perfect").
On the children's first outing to the park, they meet Bert and, despite their reservations, Mary teaches them that they must learn to look past appearances. To illustrate the point, Mary brings the park statues, including a mythological figure named Neleus, to life ("Jolly Holiday").
While Mary manages the children, other problems lie with their parents. Winifred is aware that she is somehow disappointing both her children and her husband ("Being Mrs Banks"). George, on the other hand, can't understand why she finds the role of wife and mother so difficult ("Cherry Tree Lane (reprise)"). In an effort to please her husband, Winifred sends out invitations for a smart tea party. Mrs. Brill makes the preparations, while telling an eager Robertson Ay to keep his hands off. The children inadvertently sabotage the kitchen preparations, but Mary sorts it out with a lesson ("A Spoonful of Sugar"). However, it is then revealed that none of the invitees are coming.
Mary takes the children to visit their father at the bank where he works ("Precision and Order"). There, George is busy dealing with possible investment clients: firstly, an ambitious man named Herr Von Hussler with an elaborate money-making scheme, and then a middle-class man named John Northbrook with a simple factory project. George is furious when Mary turns up with the children, but an innocent question asked by Jane (What's more important, a good man or a good idea?) makes him realise how much his values have changed ("A Man Has Dreams") since he was an idealistic young man. He then decides to accept Northbrook's project and rejects Von Hussler's, but unfortunately things take a turn for the worse afterwards.
Outside St. Paul's Cathedral, Mary introduces the children to the Bird Woman ("Feed the Birds"). Jane is suspicious of her, but Michael responds to the Bird Woman and throws crumbs for the birds. On the trip home, the children meet the enigmatic Mrs. Corry who runs a magic sweet shop that also sells words ("Supercalifragilisticexpialidocious").
The children return home in high spirits, unaware that things have gone wrong for their father. Unknown to them, George's decision to reject Von Hussler has cost the bank dearly, and he is suspended without pay. George explodes with rage at the children and they are sent to the nursery. Reacting to their father's outburst, Mary briefly points out that other members of the family are rarely there to take care of the parents, before Jane and Michael get into a fight over Jane's doll Valentine, prompting a displeased Mary to order them to bed and enchant them to sleep. The frightening consequence of Jane's anger becomes apparent moments afterwards, as Valentine and the rest of Jane and Michael's disgruntled toys come to life and join Mary in teaching them a lesson in how to take better care of their belongings and toys (formerly "Temper, Temper", now "Playing the Game").
Believing that Jane and Michael need personal growth, Mary decides to leave Cherry Tree Lane ("Chim Chim Cher-ee – Rooftop Duet"), to bring them to their senses. Her distraught charges find a note saying that perhaps they will see Mary again eventually.