Magic is a very common theme. Since both 'Harry Potter' and 'Mary Poppins' are set in magical worlds, fanfictions often explore different types of magic. For example, how Mary Poppins' magic of tidying up and her flying umbrella might compare to the spells in 'Harry Potter'.
One way to blend them creatively could be by having Mary Poppins visit Hogwarts. Maybe she could use her magical nanny skills to teach the students some unique lessons, like how to clean up a potion spill with just a snap of her fingers. She could also interact with the magical creatures there, perhaps taming a hippogriff in her own special way.
She could fit in as a mysterious visitor. For example, she might show up during a Quidditch match and use her magic to save a player from a dangerous fall. Her presence would be a shock to everyone as her magic seems so different from what they know. She could then start to interact with the characters, sharing her own brand of magic with them, like teaching Hermione how to make a room clean itself with a simple song.
One possible plot could be that Mary Poppins somehow enters the wizarding world of Harry Potter. Maybe she arrives at Hogwarts to teach a special class on manners and propriety, which the Hogwarts students desperately need. She could use her magic umbrella to fly around the castle, much to the amazement of the students. Her unique brand of magic, which is more about making the ordinary extraordinary, could clash or combine with the more traditional spell - casting magic of the wizards.
She could be a mysterious visitor. Maybe she'd show up during a difficult time at Hogwarts, like when Voldemort was causing chaos. Her positive attitude and magical abilities would surely have an impact. She might be able to cheer up the students and give them hope in the face of danger.
A great plot could be that Mary Poppins' magic umbrella is actually a powerful wizarding artifact that has been lost for centuries. When it shows up at Hogwarts, it causes all sorts of chaos as different factions try to get their hands on it. Mary Poppins then has to team up with Harry Potter and his friends to protect it. She could also teach them some new spells and charms along the way. For instance, she might show them how to use the power of song to enhance their magic, like she does in her own world. And her parrot - umbrella handle could have some secret knowledge to share about the history of the wizarding world.
It could occur during a time when Harry is on the run from Voldemort. Mary Poppins, being in the right place at the right time, might see Harry in distress. She could then use her magic to create a diversion or shield Harry from Voldemort's attacks. Since Mary Poppins has the ability to enter and exit different realms quite easily, she could whisk Harry away to a safe place, perhaps even a place outside of both the wizarding and Muggle worlds where Voldemort couldn't reach him.
Well, I found 'The Meeting of Mary Poppins and Harry Potter'. This fanfiction imagines what would happen if Mary Poppins was called upon by Dumbledore to help Harry during a particularly difficult time. It shows how Mary Poppins' no - nonsense attitude and her magical abilities blend with the world of Harry Potter. It has some really heartwarming moments as she tries to understand and assist the young wizard.
The link could be anything from character crossovers to sharing similar themes of magic and adventure. It's all up to the imagination of the fanfic writer to make it work.