The movie takes place in two time periods: the "present" of the 60's, and the past of turn-of-the- century Australia, which are seen in Travers' flashbacks to her childhood. In the present, Walt Disney has been trying for twenty years to convince a crusty Travers to allow him to make 'Mary Poppins.' That she has denied his pleas for two decades gives you some idea of her stubborn nature. Her agent finally gets her to consider it only because she is in dire financial straits. She agrees to fly to Los Angeles for a consultation, but is privately resolved to reject the contract given the least excuse to do so.
Gradually Travers becomes a bit less rigid, accepting- or at least resigning herself to the fact- that some some changes to her story are necessary, and may not even be a bad thing. In their turn, the writers realize through their (uncomfortable) interactions with Travers, that they have misinterpreted Mary Poppins' primary mission, and change the script accordingly.
Just when you think that Travers has softened enough to sign the contract, she discovers that the movie is going to contain animation, which she completely despises. This gives her an excuse to do what she had originally determined to do: pull the plug on the entire production. She returns to England, leaving Walt Disney to try to figure out how to deal with this calamity. He puzzles out what makes Travers tick, and then 'Mohammed goes to the mountain': Disney flies to London to confront her with some hard truths and settle the matter once and for all.