Harry Potter, Aunt Marge and the Harry Potter series, with all associated ideas, depicted events and settings, along with the Prisoner of Azkaban, are copyright J K Rowling. The books are published by Bloomsbury, the film series produced by Warner Bros. No infringement intended.
OK, now this is about the physics of Aunt Marge. Is it physically possible for someone to be inflated to the point of being lighter than air? Well, obviously not - it would be magic. However, if she were infinitely elastic, she would have to be over five metres in diameter. This also means that in order to be lifted by a hydrogen balloon, that would have to be at least that size, ignoring the mass of hydrogen involved.
This is based on two assumptions: Pam Ferris as Aunt Marge and the idea that Aunt Marge has a BMI of about 35, i.e. that she weighs 90 kg. Rather surprisingly, Harris tweed is quite a suitable fabric for inflation because it has a lot of give in it due to the twill weave, which skips fibres and is therefore quite loose and open compared to tabby weave.
The minister for magic claims that she was punctured and her memory altered. This seems to imply that she didn't pop but deflated relatively gracefully.
As you can see from my diagram, the whole idea is a bit far-fetched, but maybe a bit of Wingardium Leviosa was involved.